Device and method of controlling exhaust gas sensor temperature, and recording medium for exhaust gas sensor temperature control program

ABSTRACT

An apparatus for controlling temperature of an exhaust gas sensor can include a temperature estimating device for sequentially estimating the temperature of an active element based on a predetermined element temperature model, which is representative of a temperature change of the active element due to heat transfer between the active element and an exhaust gas held in contact with the active element. A heater control device can control a heater to equalize the temperature of the active element with a predetermined target temperature using an estimated value of the temperature of the active element from the temperature estimating device.

TECHNICAL FIELD

The present invention relates to an apparatus for and a method ofcontrolling the temperature of an exhaust gas sensor disposed in theexhaust passage-of an internal combustion engine, and a recording mediumstoring a program for controlling the temperature of such an exhaust gassensor.

BACKGROUND ART

Exhaust gas sensors are often disposed in the exhaust passages ofinternal combustion engines for detecting a physical quantity as to anexhaust gas component state, such as an exhaust gas componentconcentration, for the purpose of controlling the operation of theinternal combustion engine or monitoring the status of an exhaust gaspurifying system. Specifically, an exhaust gas sensor is disposed at acertain location in the exhaust gas passage and has an element sensitiveto an exhaust gas component state to be detected, the element beingpositioned for contact with an exhaust gas flowing through the exhaustpassage. For example, an air-fuel ratio sensor such as an O₂ sensor orthe like is disposed as an exhaust gas sensor upstream or downstream ofan exhaust gas purifying catalyst disposed in the exhaust passage forthe purpose of controlling the air-fuel ratio of the internal combustionengine in order to keep well the purifying ability of the catalyst.

Some air-fuel ratio sensors have a built-in heater for heating theactive element thereof for increasing the temperature of the element andactivating the element to enable the element to perform its essentialfunctions and also removing foreign matter deposited on the element. Forexample, an air-fuel ratio sensor such as an O₂ sensor or the likeusually has an electric heater for heating the active element thereof.After the internal combustion engine has started to operate, theelectric heater is energized to increase the temperature of the activeelement of the O₂ sensor to activate the active element and keep theactive element active.

As shown in FIG. 3 of the accompanying drawings, the O₂ sensor producesan output voltage Vout which changes with a large gradient with respectto a change in the air-fuel ratio of an exhaust gas, i.e., which ishighly sensitive to a change in the air-fuel ratio, only in a smallrange Δ (near a stoichiometric air-fuel ratio) of values of the air-fuelratio that is represented by an oxygen concentration in the exhaust gasto which the active element is sensitive. A change in the output voltageVout of the O₂ sensor, i.e., a gradient of the output voltage Vout withrespect to the air-fuel ratio, is smaller in air-fuel ratio ranges thatare richer and leaner than the highly sensitive range Δ. The outputcharacteristics of the O₂ sensor, i.e., the gradient of the highlysensitive range Δ, etc., vary depending on the temperature of the activeelement. When the air-fuel ratio is to be controlled using the outputvoltage from the O₂ sensor, therefore, it is desirable to keep theoutput characteristics of the O₂ sensor in a desired range as much aspossible and hence to keep the temperature of the active element of theO₂ sensor in a desired temperature range as stably as possible forbetter air-fuel ratio control.

Not only O₂ sensors but also many exhaust gas sensors have their outputcharacteristics affected by the temperature of the active element. Ifthe internal combustion engine is to be controlled using the outputsignal from the O₂ sensor, then it is preferable to keep the temperatureof the active element of the exhaust gas sensor in a desired temperaturerange as stably as possible for better engine control. When the activeelement of the exhaust gas sensor is heated to clean the active element,it is also preferable to keep the temperature of the active element ofthe exhaust gas sensor in a desired temperature range for a bettercleaning effect.

As disclosed in Japanese laid-open patent publication No. 2000-304721 bythe applicant of the present application, it is known to estimate thetemperature of the active element of an exhaust gas sensor (an air-fuelratio sensor in the publication) and control the energization of aheater (an electric heater) based on the estimated temperature forthereby keep the temperature of the active element in a desiredtemperature range to obtain appropriate output characteristics from theexhaust gas sensor. According to the disclosed arrangement, theresistance of the heater is recognized from detected values of a currentflowing through the heater and a voltage applied across the heater, andthe temperature of the active element is estimated based on the detectedresistance of the heater.

According to the disclosure of the above publication, however, since thetemperature of the active element of the exhaust gas sensor is merelyestimated based on the resistance of the heater, the reception ofthermal energy, such as heat transfer between the heater and the activeelement, is not sufficiently taken into account. Therefore, it isdifficult to accurately estimate the temperature of the active elementof the exhaust gas sensor. According to the disclosure of the abovepublication, furthermore, a duty cycle which determines the electricpower to be supplied to the heater is uniquely determined by a tablefrom an estimated value of the temperature of the active element of theexhaust gas sensor. As a result, it is difficult to control thetemperature of the active element of the exhaust gas sensor stably at adesired temperature.

The present invention has been made in view of the above background. Itis an object of the present invention to provide an apparatus for and amethod of accurately estimating the temperature of the active element ofan exhaust gas sensor or a heater, and controlling the temperature ofthe active element of the exhaust gas sensor stably at a desiredtemperature, using the estimated value of the temperature. Anotherobject of the present invention is to provide a recording medium storinga temperature control program for accurately estimating the temperatureof the active element of an exhaust gas sensor or a heater, andcontrolling the temperature of the active element of the exhaust gassensor stably at a desired temperature, using the estimated value of thetemperature.

DISCLOSURE OF THE INVENTION

An apparatus for controlling the temperature of an exhaust gas sensoraccording to the present invention is a temperature control apparatusfor an exhaust gas sensor disposed in an exhaust passage of an internalcombustion engine and having an active element for contacting an exhaustgas flowing through the exhaust passage and a heater for heating theactive element. To achieve the above object, a temperature controlapparatus according to a first aspect of the present invention ischaracterized by temperature estimating means for sequentiallyestimating the temperature of the active element based on apredetermined element temperature model which is representative of atemperature change of the active element due to heat transfer between atleast the active element and an exhaust gas held in contact with theactive element, and heater control means for controlling the heater toequalize the temperature of the active element with a predeterminedtarget temperature, using an estimated value of the temperature of theactive element from the temperature estimating means.

A method of controlling the temperature of an exhaust gas sensoraccording to the present invention is a temperature control method foran exhaust gas sensor disposed in an exhaust passage of an internalcombustion engine and having an active element for contacting an exhaustgas flowing through the exhaust passage and a heater for heating theactive element. To achieve the above object, a temperature controlmethod according to the first aspect of the present invention ischaracterized by, while sequentially estimating the temperature of theactive element based on a predetermined element temperature model whichis representative of a temperature change of the active element due toheat transfer between at least the active element and an exhaust gasheld in contact with the active element, controlling the heater toequalize the temperature of the active element with a predeterminedtarget temperature, using an estimated value of the temperature of theactive element.

A recording medium storing a temperature control program for an exhaustgas sensor according to the present invention is a recording mediumreadable by a computer and storing a temperature control program forenabling the computer to perform a process of controlling thetemperature of an exhaust gas sensor disposed in an exhaust passage ofan internal combustion engine and having an active element forcontacting an exhaust gas flowing through the exhaust passage and aheater for heating the active element. To achieve the above object, arecording medium according to the first aspect of the present inventionis characterized in that the temperature control program comprises atemperature estimating program for enabling the computer to perform aprocess of sequentially estimating the temperature of the active elementbased on a predetermined element temperature model which isrepresentative of a temperature change of the active element due to heattransfer between at least the active element and an exhaust gas held incontact with the active element, and a heater control program forenabling the computer to perform a process of controlling the heater toequalize the temperature of the active element with a predeterminedtarget temperature, using an estimated value of the temperature of theactive element.

The term “heat transfer” used in the present invention covers heattransfer through direct contact and heat transfer through air. Thisdefinition applies to other inventions of the present application.

According to the first aspect of the present invention, the elementtemperature model is representative of a temperature change of theactive element due to heat transfer between at least the active elementand an exhaust gas held in contact with the active element. Therefore,when the temperature of the active element is estimated based on theelement temperature model, the temperature of the active element can beestimated taking into account a temperature change of the active elementdue to heat transfer between the active element and the exhaust gas. Asa result, the accuracy of the estimated value of the temperature of theactive element is increased. By controlling the heater to equalize thetemperature of the active element with a predetermined targettemperature (desired temperature), using the estimated value of thetemperature of the active element, it is possible to control thetemperature of the active element stably at the target temperature.

A temperature control apparatus for an exhaust gas sensor according to asecond aspect of the present invention is characterized by temperatureestimating means for sequentially estimating the temperature of theactive element based on a predetermined element temperature model whichis representative of a temperature change of the active element due toheat transfer between at least the active element and the heater, andheater control means for controlling the heater to equalize thetemperature of the active element with a predetermined targettemperature, using an estimated value of the temperature of the activeelement from the temperature estimating means.

Similarly, a temperature control method for an exhaust gas sensoraccording to the second aspect of the present invention is characterizedby, while sequentially estimating the temperature of the active elementbased on a predetermined element temperature model which isrepresentative of a temperature change of the active element due to heattransfer between at least the active element and the heater, controllingthe heater to equalize the temperature of the active element with apredetermined target temperature, using an estimated value of thetemperature of the active element.

A recording medium storing a temperature control program for an exhaustgas sensor according to the second aspect of the present invention ischaracterized in that the temperature control program comprises atemperature estimating program for enabling the computer to perform aprocess of sequentially estimating the temperature of the active elementbased on a predetermined element temperature model which isrepresentative of a temperature change of the active element due to heattransfer between at least the active element and the heater, and aheater control program for enabling the computer to perform a process ofcontrolling the heater to equalize the temperature of the active elementwith a predetermined target temperature, using an estimated value of thetemperature of the active element.

According to the second aspect of the present invention, the elementtemperature model is representative of a temperature change of theactive element due to heat transfer between at least the active elementand the heater. Therefore, when the temperature of the active element isestimated based on the element temperature model, the temperature of theactive element can be estimated taking into account a temperature changeof the active element due to heat transfer between the active elementand the heater. As a result, the accuracy of the estimated value of thetemperature of the active element is increased. By controlling theheater to equalize the temperature of the active element with apredetermined target temperature (desired temperature), using theestimated value of the temperature of the active element, it is possibleto control the temperature of the active element stably at the targettemperature.

Both the first and second aspects of the present invention shouldpreferably be combined with respect to either one of the temperaturecontrol apparatus, the temperature control method, and the recordingmedium. In this case, in the first and second aspects of the presentinvention, the element temperature model comprises a model which isdetermined to represent, in combination, a temperature change of theactive element due to heat transfer between the active element and anexhaust gas held in contact with the active element and a temperaturechange of the active element due to heat transfer between the activeelement and the heater.

With the element temperature model thus determined, when the temperatureof the active element is estimated, the temperature of the activeelement can be estimated taking into account both of a temperaturechange of the active element due to heat transfer between the activeelement and the exhaust gas held in contact therewith and a temperaturechange of the active element due to heat transfer between the activeelement and the heater. As a result, the accuracy of the estimated valueof the temperature of the active element is further increased. Bycontrolling the heater to equalize the temperature of the active elementwith a predetermined target temperature, using the estimated value ofthe temperature of the active element, it is possible to control thetemperature of the active element more stably at the target temperature.

To achieve the above object, a temperature control apparatus for anexhaust gas sensor according to a third aspect of the present inventionis characterized by temperature estimating means for sequentiallyestimating the temperature of the active element based on apredetermined heater temperature model which is representative of atemperature change of the heater due to heat transfer between at leastthe heater and the active element, and heater control means forcontrolling the heater to equalize the temperature of the heater with apredetermined target temperature, using an estimated value of thetemperature of the heater from the temperature estimating means.

Similarly, to achieve the above object, a temperature control method foran exhaust gas sensor according to the third aspect of the presentinvention is characterized by, while sequentially estimating thetemperature of the active element based on a predetermined heatertemperature model which is representative of a temperature change of theheater due to heat transfer between at least the heater and the activeelement, controlling the heater to equalize the temperature of theheater with a predetermined target temperature, using an estimated valueof the temperature of the heater.

To achieve the above object, a recording medium storing a temperaturecontrol program for an exhaust gas sensor according to the third aspectof the present invention is characterized in that the temperaturecontrol program comprises a temperature estimating program for enablingthe computer to perform a process of sequentially estimating thetemperature of the heater based on a predetermined heater temperaturemodel which is representative of a temperature change of the heater dueto heat transfer between at least the heater and the active element, anda heater control program for enabling the computer to perform a processof controlling the heater to equalize the temperature of the heater witha predetermined target temperature, using an estimated value of thetemperature of the heater.

According to the third aspect of the present invention, the heatertemperature model is representative of a temperature change of theheater due to heat transfer between at least the heater and the activeelement. Therefore, when the temperature of the heater is estimatedbased on the heater temperature model, the temperature of the heater canbe estimated taking into account a temperature change of the heater dueto heat transfer between the heater and the active element. As a result,the accuracy of the estimated value of the temperature of the heater isincreased. By controlling the heater to equalize the temperature of theheater with a predetermined target temperature (desired temperature),using the estimated value of the temperature of the heater, it ispossible to control the temperature of the heater stably at the targettemperature. Generally, the temperature of the heater and thetemperature of the active element are highly correlated to each other ina steady state wherein their temperatures are substantially constant.Therefore, since the temperature of the heater can be controlled stablyat the target temperature as described above, the temperature of theactive element can indirectly be controlled stably at a temperaturecorresponding to the target temperature for the heater.

To achieve the above object, a temperature control apparatus for anexhaust gas sensor according to a fourth aspect of the present inventionis characterized by temperature estimating means for sequentiallyestimating the temperature of the heater based on a predetermined heatertemperature model which is representative of a temperature change of theheater due to the supply of heating energy to at least the heater, andheater control means for controlling the heater to equalize thetemperature of the heater with a predetermined target temperature, usingan estimated value of the temperature of the heater from the temperatureestimating means.

Similarly, to achieve the above object, a temperature control method foran exhaust gas sensor according to the fourth aspect of the presentinvention is characterized by, while sequentially estimating thetemperature of the heater based on a predetermined heater temperaturemodel which is representative of a temperature change of the heater dueto the supply of heating energy to at least the heater, controlling theheater to equalize the temperature of the heater with a predeterminedtarget temperature, using an estimated value of the temperature of theheater.

To achieve the above object, a recording medium storing a temperaturecontrol program for an exhaust gas sensor according to the fourth aspectof the present invention is characterized in that the temperaturecontrol program comprises a temperature estimating program for enablingthe computer to perform a process of sequentially estimating thetemperature of the heater based on a predetermined heater temperaturemodel which is representative of a temperature change of the heater dueto the supply of heating energy to at least the heater, and a heatercontrol program for enabling the computer to perform a process ofcontrolling the heater to equalize the temperature of the heater with apredetermined target temperature, using an estimated value of thetemperature of the heater.

According to the fourth aspect of the present invention, the heatertemperature model is representative of a temperature change of theheater due to the supply of heating energy to at least the heater.Therefore, when the temperature of the heater is estimated based on theheater temperature model, the temperature of the heater can be estimatedtaking into account a temperature change of the heater due to the supplyof heating energy to the heater. As a result, the accuracy of theestimated value of the temperature of the heater is increased. Bycontrolling the heater to equalize the temperature of the heater with apredetermined target temperature (desired temperature), using theestimated value of the temperature of the heater, it is possible tocontrol the temperature of the heater stably at the target temperature.Thus, the temperature of the active element can indirectly be controlledstably at a temperature corresponding to the target temperature for theheater.

Since the heater for the exhaust gas sensor is usually an electricheater, the above heating energy is usually electric power.

Both the third and fourth aspects of the present invention shouldpreferably be combined with respect to either one of the temperaturecontrol apparatus, the temperature control method, and the recordingmedium. In this case, in the third and fourth aspects of the presentinvention, the heater temperature model comprises a model which isdetermined to represent, in combination, a temperature change of theheater due to heat transfer between the heater and the active elementand a temperature change of the heater due to the supply of heatingenergy to the heater.

With the heater temperature model thus determined, when the temperatureof the heater is estimated, the temperature of the heater can beestimated taking into account both of a temperature change of the heaterdue to heat transfer between the heater and the active element, and atemperature change of the heater due to the supply of heating energy tothe heater. As a result, the accuracy of the estimated value of thetemperature of the active element is further increased, and hence thetemperature of the heater can be controlled more stably at apredetermined target temperature. Since the temperature of the heatercan further be stabilized, it is possible to control the temperature ofthe active element more stably at a temperature corresponding to thetarget temperature for the heater.

To achieve the above object, a temperature control apparatus for anexhaust gas sensor according to a fifth aspect of the present inventionis characterized by temperature estimating means for sequentiallyestimating the temperature of the active element based on apredetermined element temperature model which is representative of, incombination, a temperature change of the active element due to heattransfer between the active element and an exhaust gas held in contactwith the active element, and a temperature change of the active elementdue to heat transfer between the active element and the heater, andsequentially estimating the temperature of the heater based on apredetermined heater temperature model which is representative of, incombination, a temperature change of the heater due to heat transferbetween the heater and the active element and a temperature change ofthe heater due to the supply of heating energy to the heater, and heatercontrol means for controlling the heater to equalize the temperature ofthe active element with a predetermined target temperature, using anestimated value of the temperature of the active element and anestimated value of the temperature of the heater from the temperatureestimating means.

Similarly, to achieve the above object, a temperature control method foran exhaust gas sensor according to the fifth aspect of the presentinvention is characterized by sequentially estimating the temperature ofthe active element based on a predetermined element temperature modelwhich is representative of, in combination, a temperature change of theactive element due to heat transfer between the active element and anexhaust gas held in contact with the active element, and a temperaturechange of the active element due to heat transfer between the activeelement and the heater, and sequentially estimating the temperature ofthe heater based on a predetermined heater temperature model which isrepresentative of, in combination, a temperature change of the heaterdue to heat transfer between the heater and the active element and atemperature change of the heater due to the supply of heating energy tothe heater, and controlling the heater to equalize the temperature ofthe active element with a predetermined target temperature, using anestimated value of the temperature of the active element and anestimated value of the temperature of the heater while estimating thetemperature of the active element and the temperature of the heater.

To achieve the above object, a recording medium storing a temperaturecontrol program for an exhaust gas sensor according to the fifth aspectof the present invention is characterized in that the temperaturecontrol program comprises a temperature estimating program for enablingthe computer to perform a process of sequentially estimating thetemperature of the active element based on a predetermined elementtemperature model which is representative of, in combination, atemperature change of the active element due to heat transfer betweenthe active element and an exhaust gas held in contact with the activeelement, and a temperature change of the active element due to heattransfer between the active element and the heater, and sequentiallyestimating the temperature of the heater based on a predetermined heatertemperature model which is representative of, in combination, atemperature change of the heater due to heat transfer between the heaterand the active element and a temperature change of the heater due to thesupply of heating energy to the heater, and a heater control program forenabling the computer to perform a process of controlling the heater toequalize the temperature of the active element with a predeterminedtarget temperature, using an estimated value of the temperature of theactive element and an estimated value of the temperature of the heater.

According to the fifth aspect of the present invention, since thetemperature of the active element is estimated based on the elementtemperature model according to a combination of the first and secondaspects of the present invention, and the temperature of the heater isestimated based on the heater temperature model according to acombination of the third and fourth aspects of the present invention. Bycontrolling the heater to equalize the temperature of the active elementwith a predetermined target temperature (desired temperature), usingboth the estimated value of the temperature of the active element andthe estimated value of the temperature of the heater, it is possible tocontrol the temperature of the active element effectively and stably atthe target temperature.

Specifically, when the heater is controlled to equalize the temperatureof the active element with the predetermined target temperature,inasmuch as the heater can be controlled taking into account not onlythe estimated value of the temperature of the active element as acontrolled variable, but also the estimated value of the temperature ofthe heater which affects the temperature of the active element, thestability of a process of controlling the temperature of the activeelement at the target temperature can be increased.

According to the fifth aspect described above, the heater is controlledto equalize the temperature of the active element with the predeterminedtarget temperature. Even when the heater is controlled to equalize thetemperature of the heater with the predetermined target temperatureaccording to the third and fourth aspects described above, it ispossible to apply a process which is the same as the fifth aspect asdescribed below.

To achieve the above object, a temperature control apparatus for anexhaust gas sensor according to a sixth aspect of the present inventionis characterized by temperature estimating means for sequentiallyestimating the temperature of the active element based on apredetermined element temperature model which is representative of, incombination, a temperature change of the active element due to heattransfer between the active element and an exhaust gas held in contactwith the active element, and a temperature change of the active elementdue to heat transfer between the active element and the heater, andsequentially estimating the temperature of the heater based on apredetermined heater temperature model which is representative of, incombination, a temperature change of the heater due to heat transferbetween the heater and the active element and a temperature change ofthe heater due to the supply of heating energy to the heater, and heatercontrol means for controlling the heater to equalize the temperature ofthe heater with a predetermined target temperature, using an estimatedvalue of the temperature of the active element and an estimated value ofthe temperature of the heater from the temperature estimating means.

Similarly, to achieve the above object, a temperature control method foran exhaust gas sensor according to the sixth aspect of the presentinvention is characterized by sequentially estimating the temperature ofthe active element based on a predetermined element temperature modelwhich is representative of, in combination, a temperature change of theactive element due to heat transfer between the active element and anexhaust gas held in contact with the active element, and a temperaturechange of the active element due to heat transfer between the activeelement and the heater, and sequentially estimating the temperature ofthe heater based on a predetermined heater temperature model which isrepresentative of, in combination, a temperature change of the heaterdue to heat transfer between the heater and the active element and atemperature change of the heater due to the supply of heating energy tothe heater, and controlling the heater to equalize the temperature ofthe heater with a predetermined target temperature, using an estimatedvalue of the temperature of the active element and an estimated value ofthe temperature of the heater while estimating the temperature of theactive element and the temperature of the heater.

To achieve the above object, a recording medium storing a temperaturecontrol program for an exhaust gas sensor according to the sixth aspectof the present invention is characterized in that the temperaturecontrol program comprises a temperature estimating program for enablingthe computer to perform a process of sequentially estimating thetemperature of the active element based on a predetermined elementtemperature model which is representative of, in combination, atemperature change of the active element due to heat transfer betweenthe active element and an exhaust gas held in contact with the activeelement, and a temperature change of the active element due to heattransfer between the active element and the heater, and sequentiallyestimating the temperature of the heater based on a predetermined heatertemperature model which is representative of, in combination, atemperature change of the heater due to heat transfer between the heaterand the active element and a temperature change of the heater due to thesupply of heating energy to the heater, and a heater control program forenabling the computer to perform a process of controlling the heater toequalize the temperature of the heater with a predetermined targettemperature, using an estimated value of the temperature of the activeelement and an estimated value of the temperature of the heater.

According to the sixth aspect of the present invention, since thetemperatures of the active element and the heater are estimated inexactly the same manner as with the fifth aspect described, theestimated values of the temperatures thereof are accurately determined.By controlling the heater to equalize the temperature of the heater withthe predetermined target temperature (desired temperature), using boththe estimated value of the temperature of the active element and theestimated value of the temperature of the heater, the temperature of theheater can be controlled effectively and stably at the targettemperature.

Specifically, when the heater is controlled to equalize the temperatureof the heater with the predetermined target temperature, inasmuch as theheater can be controlled taking into account not only the estimatedvalue of the temperature of the heater as a controlled variable, butalso the estimated value of the temperature of the active element whichaffects the temperature of the heater, the stability of a process ofcontrolling the temperature of the heater at the target temperature canbe increased. Hence, the temperature of the active element can becontrolled with high stability at a temperature corresponding to thetarget temperature for the heater.

In the first aspect of the present invention which uses the elementtemperature model taking into account heat transfer between the activeelement and the exhaust gas for estimating the temperature of the activeelement, the element temperature model should preferably comprise amodel which is representative of a change per predetermined time in thetemperature of the active element as including a temperature changecomponent depending on the difference between at least the temperatureof the active element and the temperature of the exhaust gas held incontact with the active element. In the control apparatus according tothe first aspect, the temperature estimating means should preferablysequentially estimate a temperature change of the active element basedon the element temperature model, and accumulatively add an estimatedvalue of the temperature change to an initial value which is set whenthe internal combustion engine starts to operate, thereby estimating thetemperature of the active element. Similarly, in the control methodaccording to the second aspect, while sequentially estimating atemperature change of the active element based on the elementtemperature model, an estimated value of the temperature change shouldpreferably be accumulatively added to an initial value which is set whenthe internal combustion engine starts to operate, thereby estimating thetemperature of the active element. In the recording medium according tothe first aspect, the temperature estimating program should preferablycomprise a program for enabling the computer to perform a process ofsequentially estimating a temperature change of the active element basedon the element temperature model, and accumulatively adding an estimatedvalue of the temperature change to an initial value which is set whenthe internal combustion engine starts to operate, thereby estimating thetemperature of the active element.

In the element temperature model according to the first aspect, atemperature change component (a temperature change component perpredetermined time) depending on the difference between the temperatureof the active element and the temperature of the exhaust gas held incontact with the active element means a temperature change component ofthe active element depending on heat transfer between the active elementand the exhaust gas. Therefore, based on the element temperature modelincluding the temperature change component, an estimated value of thetemperature change per predetermined time of the active element can bedetermined in a manner appropriately taking into account heat transferbetween the active element and the exhaust gas. It is possible toaccurately determine the estimated value of the temperature of theactive element by accumulatively adding the estimated value of thetemperature change to an initial value which is set when the internalcombustion engine starts to operate (a predicted value of thetemperature of the active element at the time the internal combustionengine starts to operate). Thus, the heater can well be controlled toequalize the temperature of the active element with the predeterminedtarget temperature for thereby increasing the stability of thetemperature of the active element.

The above process of sequentially estimating the temperature change perpredetermined time of the active element and accumulatively adding theestimated value to the initial value can also be applied to the secondaspect of the present invention which uses the element temperature modeltaking into account heat transfer between the active element and theheater. Specifically, in the second aspect of the present invention, theelement temperature model should preferably comprise a model which isrepresentative of a change per predetermined time in the temperature ofthe active element as including a temperature change component dependingon the difference between at least the temperature of the active elementand the temperature of the heater. In the control apparatus according tothe second aspect of the present invention, the temperature estimatingmeans should preferably sequentially estimate a temperature change ofthe active element based on the element temperature model, andaccumulatively add an estimated value of the temperature change to aninitial value which is set when the internal combustion engine starts tooperate, thereby estimating the temperature of the active element. Inthe control method according to the second aspect, while sequentiallyestimating a temperature change of the active element based on theelement temperature model, an estimated value of the temperature changeshould preferably be accumulatively added to an initial value which isset when the internal combustion engine starts to operate, therebyestimating the temperature of the active element. In the recordingmedium according to the second aspect, the temperature estimatingprogram should preferably comprise a program for enabling the computerto perform a process of sequentially estimating a temperature change ofthe active element based on the element temperature model, andaccumulatively adding an estimated value of the temperature change to aninitial value which is set when the internal combustion engine starts tooperate, thereby estimating the temperature of the active element.

In the element temperature model according to the second aspect, atemperature change component depending on the difference between thetemperature of the active element and the temperature of the heatermeans a temperature change component (a temperature change component perpredetermined time) of the active element depending on heat transferbetween the active element and the heater. Therefore, based on theelement temperature model including the temperature change component, anestimated value of the temperature change per predetermined time of theactive element can be determined in a manner appropriately taking intoaccount heat transfer between the active element and the heater. It ispossible to accurately determine the estimated value of the temperatureof the active element by accumulatively adding the estimated value ofthe temperature change to an initial value which is set when theinternal combustion engine starts to operate (a predicted value of thetemperature of the active element at the time the internal combustionengine starts to operate). Thus, the heater can well be controlled toequalize the temperature of the active element with the predeterminedtarget temperature for thereby increasing the stability of thetemperature of the active element.

The above process of sequentially estimating the temperature change perpredetermined time of the active element and accumulatively adding theestimated value to the initial value can also be applied to acombination of the element temperature model according to the firstaspect and the element temperature model according to the second aspect.In this case, the element temperature model should preferably comprise amodel which is representative of a change per predetermined time in thetemperature of the active element as including a temperature changecomponent depending on the difference between the temperature of theactive element and the temperature of the exhaust gas held in contactwith the active element, and a temperature change component depending onthe difference between the temperature of the active element and thetemperature of the heater. In the temperature control apparatusaccording to a combination of the first and second aspects, thetemperature estimating means should preferably sequentially estimate atemperature change of the active element based on the elementtemperature model, and accumulatively add an estimated value of thetemperature change to an initial value which is set when the internalcombustion engine starts to operate, thereby estimating the temperatureof the active element. In the temperature control method according to acombination of the first and second aspects, while sequentiallyestimating a temperature change of the active element based on theelement temperature model, an estimated value of the temperature changeshould preferably be accumulatively added to an initial value which isset when the internal combustion engine starts to operate, therebyestimating the temperature of the active element. In the recordingmedium according to a combination of the first and second aspects, thetemperature estimating program should preferably comprise a program forenabling the computer to perform a process of sequentially estimating atemperature change of the active element based on the elementtemperature model, and accumulatively adding an estimated value of thetemperature change to an initial value which is set when the internalcombustion engine starts to operate, thereby estimating the temperatureof the active element.

With the above arrangement, an estimated value of the temperature changeper predetermined time of the active element can accurately bedetermined in a manner appropriately taking into account two types ofheat transfer, i.e., heat transfer between the active element and theexhaust gas and heat transfer between the active element and the heater.It is possible to more accurately determine the estimated value of thetemperature of the active element by accumulatively adding the estimatedvalue of the temperature change to an initial value which is set whenthe internal combustion engine starts to operate (a predicted value ofthe temperature of the active element at the time the internalcombustion engine starts to operate). Thus, the heater can well becontrolled to equalize the temperature of the active element with thepredetermined target temperature for thereby further increasing thestability of the temperature of the active element.

In the first and second aspects, when the temperature of the activeelement is estimated using the element temperature model which isrepresentative of a temperature change per predetermined time of theactive element, as described above, the temperature of the exhaust gaswhich is required to determine a temperature change per predeterminedtime of the active element based on the element temperature model, andthe temperature of the heater are directly detected using temperaturesensors (latest detected values). However, values estimated fromappropriate parameters (latest estimated values) may be used as thosetemperatures.

The above process of using the element temperature model which isrepresentative of a temperature change per predetermined time of theactive element in order to estimate the temperature of the activeelement should preferably be performed also in the third or fourthaspect of the present invention which estimates the temperature of theheater or a combination of the third and fourth aspects. Specifically,in the third aspect described above, the heater temperature model shouldpreferably comprise a model which is representative of a change perpredetermined time in the temperature of the heater as including atemperature change component depending on the difference between thetemperature of the heater and the temperature of the active element. Inthe control apparatus according to the third aspect, the temperatureestimating means should preferably sequentially estimate a temperaturechange of the heater based on the heater temperature model, andaccumulatively add an estimated value of the temperature change to aninitial value which is set when the internal combustion engine starts tooperate, thereby estimating the temperature of the heater. In thecontrol method according to the third aspect, while sequentiallyestimating a temperature change of the heater based on the heatertemperature model, an estimated value of the temperature change shouldpreferably be accumulatively added to an initial value which is set whenthe internal combustion engine starts to operate, thereby estimating thetemperature of the heater. In the recording medium according to thethird aspect, the temperature estimating program should preferablycomprise a program for enabling the computer to perform a process ofsequentially estimating a temperature change of the heater based on theheater temperature model, and accumulatively adding an estimated valueof the temperature change to an initial value which is set when theinternal combustion engine starts to operate, thereby estimating thetemperature of the heater.

In the fourth aspect described above, the heater temperature modelshould preferably comprise a model which is representative of a changeper predetermined time in the temperature of the heater as including atemperature change component depending on an amount of heating energysupplied to the heater. In the temperature control apparatus accordingto the fourth aspect, the temperature estimating means sequentiallyestimates a temperature change of the heater based on the heatertemperature model, and accumulatively adds an estimated value of thetemperature change to an initial value which is set when the internalcombustion engine starts to operate, thereby estimating the temperatureof the heater. In the temperature control method according to the fourthaspect, while sequentially estimating a temperature change of the heaterbased on the heater temperature model, an estimated value of thetemperature change should preferably be accumulatively added to aninitial value which is set when the internal combustion engine starts tooperate, thereby estimating the temperature of the heater. In therecording medium according to the fourth aspect, the temperatureestimating program should preferably comprise a program for enabling thecomputer to perform a process of sequentially estimating a temperaturechange of the heater based on the heater temperature model, andaccumulatively adds an estimated value of the temperature change to aninitial value which is set when the internal combustion engine starts tooperate, thereby estimating the temperature of the heater.

If the third aspect and the fourth aspect are combined with each other,then the heater temperature model should preferably comprise a modelwhich is representative of a change per predetermined time in thetemperature of the heater as including a temperature change componentdepending on the difference between the temperature of the heater andthe temperature of the active element, and a temperature changecomponent depending on an amount of heating energy supplied to theheater. In the temperature control apparatus according to a combinationof the third and fourth aspects, the temperature estimating means shouldpreferably sequentially estimate a temperature change of the heaterbased on the heater temperature model, and accumulatively add anestimated value of the temperature change to an initial value which isset when the internal combustion engine starts to operate, therebyestimating the temperature of the heater. In the temperature controlmethod according to a combination of the third and fourth aspects, whilesequentially estimating a temperature change of the heater based on theheater temperature model, an estimated value of the temperature changeshould preferably be accumulatively added to an initial value which isset when the internal combustion engine starts to operate, therebyestimating the temperature of the heater. In the recording mediumaccording to a combination of the third and fourth aspects, thetemperature estimating program should preferably comprise a program forenabling the computer to perform a process of sequentially estimating atemperature change of the heater based on the heater temperature model,and accumulatively adding an estimated value of the temperature changeto an initial value which is set when the internal combustion enginestarts to operate, thereby estimating the temperature of the heater.

If the heater temperature model which is representative of a temperaturechange per predetermined time of the heater is used, then in the heatertemperature model, a temperature change component (a temperature changecomponent per predetermined time) depending on the difference betweenthe temperature of the heater and the temperature of the active elementmeans a temperature change component of the heater depending on heattransfer between the heater and the active element. Furthermore, atemperature change component of the heater depending on the amount ofheating energy supplied to the heater means a temperature changecomponent (a temperature change component per predetermined time) of theheater depending on the supply of heating energy to the heater.

In the third aspect described above, therefore, an estimated value of atemperature change per predetermined time of the heater can bedetermined in a manner appropriately taking into account heat transferbetween the heater and the active element based on the heatertemperature model. Similarly, in the fourth aspect described above, anestimated value of a temperature change per predetermined time of theheater can be determined in a manner appropriately taking into accountthe supply of heating energy to the heater. In particular, if the thirdand fourth aspects are combined with each other, then an estimated valueof a temperature change per predetermined time of the heater can bedetermined more accurately in a manner appropriately taking into accountboth heat transfer between the heater and the active element and thesupply of heating energy to the heater. The estimated value of thetemperature of the heater can thus be determined accurately byaccumulatively adding the estimated value of the temperature change toan initial value which is set when the internal combustion engine startsto operate (a predicted value of the temperature of the heater at thetime the internal combustion engine starts to operate). Thus, the heatercan well be controlled to equalize the temperature of the heater withthe predetermined target temperature for thereby increasing thestability of the temperature of the active element.

The temperature of the active element which is required to determine atemperature change of the heater per predetermined time based on theheater temperature model is directly detected using a temperature sensor(latest detected value). However, a value estimated from an appropriateparameter (latest estimated value) may be used as the temperature of theactive element. The amount of heating energy supplied to the heater,which is required to determine a temperature change of the heater, maybe a value (latest value) of a control input (controlled variable) thatis generated in controlling the heater as a quantity for determining theamount of heating energy supplied to the heater. Alternatively, if theheater is an electric heater, then a current flowing through and avoltage applied to the heater, for example, may be detected, andelectric power supplied to the heater which is grasped from detectedvalues (latest detected values) may be used as a quantity fordetermining the amount of heating energy supplied to the heater.

In the fifth and sixth aspects of the present invention which estimateboth the temperature of the active element and the temperature of theheater, it is preferably to establish an element temperature model and aheater temperature model for estimating the temperature of the activeelement and the temperature of the heater in the same manner asdescribed above. Specifically, in either one of the fifth and sixthaspects, the element temperature model should preferably comprise amodel which is representative of a change per predetermined time in thetemperature of the active element as including a temperature changecomponent depending on the difference between the temperature of theactive element and the temperature of the exhaust gas held in contactwith the active element, a temperature change component depending on thedifference between the temperature of the active element and thetemperature of the heater, and the heater temperature model shouldpreferably comprise a model which is representative of a change perpredetermined time in the temperature of the heater as including atemperature change component depending on the difference between thetemperature of the heater and the temperature of the active element, anda temperature change component depending on an amount of heating energysupplied to the heater. In the temperature control apparatus accordingto the fifth or sixth aspect, the temperature estimating means shouldpreferably sequentially estimate a temperature change of the activeelement based on the element temperature model, and accumulatively addan estimated value of the temperature change to an initial value of thetemperature of the active element which is set when the internalcombustion engine starts to operate, thereby estimating the temperatureof the active element, and should preferably sequentially estimate atemperature change of the heater based on the heater temperature model,and accumulatively adds an estimated value of the temperature change toan initial value of the temperature of the heater which is set when theinternal combustion engine starts to operate, thereby estimating thetemperature of the heater. In the temperature control method accordingto the fifth or sixth aspect, while sequentially estimating atemperature change of the active element based on the elementtemperature model, an estimated value of the temperature change shouldpreferably be accumulatively added to an initial value of thetemperature of the active element which is set when the internalcombustion engine starts to operate, thereby estimating the temperatureof the active element, and while sequentially estimating a temperaturechange of the heater based on the heater temperature model, an estimatedvalue of the temperature change should preferably be accumulativelyadded to an initial value of the temperature of the heater which is setwhen the internal combustion engine starts to operate, therebyestimating the temperature of the heater. In the recording mediumaccording to the fifth or sixth aspect, the temperature estimatingprogram should preferably comprise a program for enabling the computerto perform a process of sequentially estimating a temperature change ofthe active element based on the element temperature model, andaccumulatively adding an estimated value of the temperature change to aninitial value of the temperature of the active element which is set whenthe internal combustion engine starts to operate, thereby estimating thetemperature of the active element, and sequentially estimating atemperature change of the heater based on the heater temperature model,and accumulatively adding an estimated value of the temperature changeto an initial value of the temperature of the heater which is set whenthe internal combustion engine starts to operate, thereby estimating thetemperature of the heater.

With the above arrangement, temperature changes per predetermined timeof the active element and the heater can accurately be determined in amanner appropriately taking into account the application of heat to theactive element and the heater. By accumulatively adding an estimatedvalue of the temperature change per predetermined time of the activeelement to an initial value of the temperature of the active elementwhich is set when the internal combustion engine starts to operate (apredicted value of the temperature of the active element at the time theinternal combustion engine starts to operate), it is possible toaccurately determine the estimated value of the temperature of theactive element. Likewise, by accumulatively adding an estimated value ofthe temperature change per predetermined time of the heater to aninitial value of the temperature of the active element which is set whenthe internal combustion engine starts to operate (a predicted value ofthe temperature of the heater at the time the internal combustion enginestarts to operate), it is possible to accurately determine the estimatedvalue of the temperature of the heater. Thus, the heater can well becontrolled to equalize the temperature of the active element or theheater with the target temperature for thereby effectively increasingthe stability of the temperature of the active element.

In this case, the temperature of the heater which is required todetermine a temperature change of the active element based on theelement temperature model may be an estimated value (latest value) basedon the heater temperature model. Similarly, the temperature of theactive element which is required to determine a temperature change ofthe heater based on the heater temperature model may be an estimatedvalue (latest value) based on the element temperature model. Thetemperature of the exhaust gas which is required to determine atemperature change of the active element based on the elementtemperature model may be either one of detected and estimated values, aswith the first aspect described above. The amount of heating energysupplied to the heater, which is required to determine a temperaturechange of the heater based on the heater temperature model, may be acontrol input that is generated in controlling the heater as a quantityfor determining the amount of heating energy supplied to the heater orelectric power supplied to the heater which is grasped from detectedvalues of a current flowing through and a voltage applied to the heater.

According to the present invention which estimates the temperature ofthe active element and the temperature of the heater by accumulativelyadding an estimated value of a temperature change per predetermined timeto an initial value, in either one of the first through sixth aspectsdescribed above, the initial value should preferably be set depending onthe atmospheric temperature and/or the temperature of the internalcombustion engine at least when the internal combustion engine starts tooperate. With this arrangement, an initial value of the temperature ofthe active element and the temperature of the heater when the internalcombustion engine starts to operate can appropriately be established. Ifthe period of time in which the internal combustion engine is at rest islong before it starts to operate, then it is preferable to use theatmospheric temperature as the initial value. If the period of time inwhich the internal combustion engine is at rest is short, then it ispreferable to use the engine temperature as the initial value.

In the temperature control apparatus according to the first aspect, thesecond aspect, or a combination of the first and second aspects of thepresent invention, the heater control means should preferablysequentially generate a control input which determines an amount ofheating energy supplied to the heater, depending on at least theestimated value of the temperature of the active element from thetemperature estimating means, and control the heater depending on thecontrol input. In the temperature control method according to the firstaspect, the second aspect, or a combination of the first and secondaspects of the present invention, while sequentially generating acontrol input which determines an amount of heating energy supplied tothe heater, depending on at least the estimated value of the temperatureof the active element, the heater should preferably be controlleddepending on the control input. In the recording medium according to thefirst aspect, the second aspect, or a combination of the first andsecond aspects of the present invention, the heater control programshould preferably comprise a program for enabling the computer toperform a process of sequentially generating a control input whichdetermines an amount of heating energy supplied to the heater, dependingon at least the estimated value of the temperature of the activeelement, and controlling the heater depending on the control input.

The control input thus generated includes a feedback component dependingon the estimated value of the temperature of the active element as acontrol quantity. Therefore, the heater is controlled according to afeedback control process. The heater can thus well be controlled tocontrol the temperature of the active element at the predeterminedtarget temperature, thereby appropriately keeping the stability of thetemperature of the active element.

In the temperature control apparatus according to the third aspect, thefourth aspect, or a combination of the third and fourth aspects of thepresent invention, the heater control means should preferablysequentially generate a control input which determines an amount ofheating energy supplied to the heater, depending on at least theestimated value of the temperature of the heater from the temperatureestimating means, and control the heater depending on the control input.In the temperature control method according to the third aspect, thefourth aspect., or a combination of the third and fourth aspects of thepresent invention, while sequentially generating a control input whichdetermines an amount of heating energy supplied to the heater, dependingon at least the estimated value of the temperature of the heater, theheater should preferably be controlled depending on the control input.In the recording medium according to the third aspect, the fourthaspect, or a combination of the third and fourth aspects of the presentinvention, the heater control program should preferably comprise aprogram for enabling the computer to perform a process of sequentiallygenerating a control input which determines an amount of heating energysupplied to the heater, depending on at least the estimated value of thetemperature of the heater, and controlling the heater depending on thecontrol input.

The control input thus generated includes a feedback component dependingon the estimated value of the temperature of the heater as a controlquantity. Therefore, the heater is controlled according to a feedbackcontrol process. The heater can thus well be controlled to control thetemperature of the heater at the predetermined target temperature,thereby appropriately keeping the stability of the temperature of theactive element.

In the temperature control apparatus according to the fifth aspect orthe sixth aspect of the present invention, the heater control meansshould preferably sequentially generate a control input which determinesan amount of heating energy supplied to the heater by adding an inputcomponent depending on at least the estimated value of the temperatureof the active element from the temperature estimating means and theestimated value of the temperature of the heater from the temperatureestimating means, and control the heater depending on the control input.In the temperature control method according to the fifth aspect or thesixth aspect of the present invention, while sequentially generating acontrol input which determines an amount of heating energy supplied tothe heater by adding an input component depending on at least theestimated value of the temperature of the active element and theestimated value of the temperature of the heater, the heater shouldpreferably be controlled depending on the control input. In therecording medium according to the fifth aspect or the sixth aspect ofthe present invention, the heater control program should preferablycomprise a program for enabling the computer to perform a process ofsequentially generating a control input which determines an amount ofheating energy supplied to the heater by adding an input componentdepending on at least the estimated value of the temperature of theactive element and the estimated value of the temperature of the heater,and controlling the heater depending on the control input.

In the fifth aspect, the control input thus generated includes afeedback component depending on the estimated value of the temperatureof the active element as a control quantity and, in addition, a controlinput component depending on the estimated value of the temperature ofthe heater. In the sixth aspect, the control input includes a controlinput component depending on the estimated value of the temperature ofthe active element, in addition to a feedback component depending on theestimated value of the temperature of the heater as a control quantity.As a result, the stability of a process of controlling the heater forequalizing the temperature of the active element or the temperature ofthe heater at the target temperature can further be increased. Hence,the stability of the temperature of the active element is moreeffectively maintained.

According to the present invention, the exhaust gas sensor may comprisean O₂ sensor disposed downstream of a catalytic converter for purifyingthe exhaust gas, for example. If the air-fuel ratio of the exhaust gasto keep the output voltage of the O₂ sensor at a predetermined level inorder for the catalytic converter to perform its desired exhaust gaspurifying capability, the temperature of the active element of the O₂sensor should preferably be controlled at a temperature equal to orhigher than 750° C. (e.g., 800° C.). In this case, when the heater is tobe controlled with a target temperature determined for the activeelement, the target temperature may be set to a temperature equal to orhigher than 750° C. (e.g., 800° C.). When the heater is to be controlledwith a target temperature determined for the heater, the targettemperature may be set to a temperature equal to or higher than 850° C.(e.g., 900° C.).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an apparatus according to a firstembodiment of the present invention;

FIG. 2 is a fragmentary cross-sectional view showing a structure of anO₂ sensor (exhaust gas sensor) in the apparatus shown in FIG. 1;

FIG. 3 is a graph illustrative of the output characteristics of the O₂sensor shown in FIG. 2;

FIG. 4 is a block diagram showing a functional arrangement of a sensortemperature control means in the apparatus shown in FIG. 1;

FIG. 5 is a cross-sectional view showing a processing operation of anexhaust temperature observer in the sensor temperature control meansshown in FIG. 4;

FIG. 6 is a block diagram showing a functional arrangement of theexhaust temperature observer in the sensor temperature control meansshown in FIG. 4;

FIG. 7 is a block diagram showing a functional arrangement of a heatercontroller in the sensor temperature control means shown in FIG. 4;

FIG. 8 is a flowchart of an overall processing sequence of the sensortemperature control means in the apparatus shown in FIG. 1;

FIGS. 9 through 11 are flowcharts of subroutines of the flowchart shownin FIG. 8; and

FIG. 12 is a block diagram showing a functional arrangement of a sensortemperature control means in an apparatus according to a secondembodiment of the present invention.

BEST MODE FOR CARRYING OUT THE INVENTION

A first embodiment of the present invention will be described below withreference to FIGS. 1 through 11. FIG. 1 shows in block form an overallarrangement of the apparatus according to the first embodiment of thepresent invention. In FIG. 1, an engine (an internal combustion engine)1 mounted on an automobile, a hybrid vehicle, or the like combusts amixture of fuel and air and generates an exhaust gas, which isdischarged into the atmosphere through an exhaust passage 3communicating with an exhaust port 2 of the engine 1. The exhaustpassage 3 incorporates therein two catalytic converters 4, 5 disposedsuccessively downstream for purifying the exhaust gas emitted from theengine 1 and flowing through the exhaust passage 3. The exhaust passage3 includes a section upstream of the catalytic converter 4 (between theexhaust port 2 and the catalytic converter 4), a section between thecatalytic converters 4, 5, and a section downstream of the catalyticconverter 5. These sections of the exhaust passage 3 are provided byrespective exhaust pipes 6 a, 6 b, 6 c each in the form of a tubularpassage-defining member.

Each of the catalytic converters 4, 5 contains a catalyst 7 (three-waycatalyst in the present embodiment). The catalyst 7 has apassage-defining honeycomb structure and allows the exhaust gas to flowtherethrough. Though the catalytic converters 4, 5 may be of a unitarystructure with two catalytic beds, each comprising a three-way catalyst,disposed respectively in upstream and downstream regions thereof.

In the present embodiment, the air-fuel ratio in the exhaust gas emittedfrom the engine 1 is controlled in order for the upstream catalyticconverter 4, in particular, to have a good exhaust gas purifyingcapability (the ability of the catalytic converter 4 to purify CO, HC,and NOx). For controlling the air-fuel ratio in the exhaust gas, an O₂sensor 8 is mounted on the exhaust passage 3 between the catalyticconverters 4, 5, i.e., on the exhaust passage defined by the exhaustpipe 6 b, and a wide-range air-fuel ratio sensor 9 is mounted on theexhaust passage 3 upstream of the catalytic converter 4, i.e., on theexhaust passage defined by the exhaust pipe 6 a.

The O₂ sensor 8 corresponds to an exhaust gas sensor according to thepresent invention. Basic structural details and characteristics of theO₂ sensor 8 will be described below. As shown in FIG. 2, the O₂ sensor 8has an active element 10 (sensitive element) in the form of a hollowbottomed cylinder made primarily of a solid electrolyte permeable tooxygen ions, e.g., stabilized zirconia (ZrO₂+Y₂O₃). The active element10 has outer and inner surfaces coated with porous platinum electrodes11, 12, respectively. The O₂ sensor 8 also has a rod-shaped ceramicheater 13 inserted as an electric heater into the active element 10 forheating the active element 10 for activation and controlling thetemperature of the active element 10. The active element 10 is filledwith air containing oxygen at a constant concentration, i.e., under aconstant partial pressure, in a space around the ceramic heater 13. TheO₂ sensor 8 is placed in a sensor casing 14 mounted on the exhaust pipe6 b such that the tip end of the active element 10 has its outer surfacepositioned in contact with the exhaust gas flowing in the exhaust pipe 6b.

The tip end of the active element 10 is covered with a tubular protector15 which protects the active element 10 against the impingement offoreign matter thereon. The tip end of the active element 10 which ispositioned in the exhaust pipe 6 b contacts the exhaust gas through aplurality of holes (not shown) defined in the protector 15.

The O₂ sensor 8 thus constructed operates as follows: An electromotiveforce depending on the concentration of oxygen in the exhaust gas isgenerated between the platinum electrodes 11, 12 based on the differencebetween the concentration of oxygen in the exhaust gas which is broughtinto contact with the outer surface of the tip end of the active element10 and the concentration of oxygen in the air in the active element 10.The generated electromotive force is amplified by an amplifier (notshown), and then produced as the output voltage Vout from the O₂ sensor8.

The output voltage Vout of the O₂ sensor 8 has characteristics (outputcharacteristics) with respect to the concentration of oxygen in theexhaust gas or the air-fuel ratio in the exhaust gas which is recognizedfrom the concentration of oxygen, as represented by a solid-line curve“a” (so-called “Z curve”) in FIG. 3. The solid-line curve “a” representsthe output characteristics of the O₂ sensor 8 when the temperature ofthe active element 10 is 800° C. The relationship between thetemperature of the active element 10 and the output characteristics ofthe O₂ sensor 8 will be described later on.

As indicated by the curve “a” in FIG. 3, the output characteristics ofthe O₂ sensor 8 are generally of such a nature that the output voltageVout changes substantially linearly with high sensitivity with respectto the air-fuel ratio of the exhaust gas only when the air-fuel ratiorepresented by the concentration of oxygen in the exhaust gas is presentin a narrow air-fuel ratio range Δ near a stoichiometric air-fuel ratio.In the air-fuel ratio range Δ (hereinafter referred to as“high-sensitivity air-fuel ratio range Δ”), the gradient of a change inthe output voltage Vout with respect to a change in the air-fuel ratio,i.e., the gradient of the curve of the output characteristics of the O₂sensor 8, is large. In an air-fuel ratio range richer than thehigh-sensitivity air-fuel ratio range Δ and an air-fuel ratio rangeleaner than the high-sensitivity air-fuel ratio range Δ, the gradient ofa change in the output voltage Vout with respect to a change in theair-fuel ratio, i.e., the gradient of the curve of the outputcharacteristics of the O₂ sensor 8, is smaller.

The wide-range air-fuel ratio sensor 9, which will not be described indetail below, comprises an air-fuel ratio sensor disclosed in Japaneselaid-open patent publication No. 4-369471 by the applicant of thepresent application, for example. The wide-range air-fuel ratio sensor 9is a sensor for generating an output voltage KACT which changes linearlywith respect to the air-fuel ratio in the exhaust gas in an air-fuelratio range wider than the O₂ sensor 8. The output voltage Vout of theO₂ sensor 8 and the output voltage KACT of the wide-range air-fuel ratiosensor 9 will hereinafter be referred to as “output Vout” and “outputKACT”, respectively.

As shown in FIG. 1, the apparatus according to the present embodimentalso has a control unit 16 for controlling the air-fuel ratio in theexhaust gas and controlling the temperature of the active element 10 ofthe O₂ sensor 8. The control unit 16 comprises a microcomputer includinga CPU, a RAM, and a ROM (not shown). For carrying out a control processto be described later on, the control unit 16 is supplied with theoutputs Vout and KACT from the O₂ sensor 8 and the wide-range air-fuelratio sensor 9, and also with data representing the rotational speed NEof the engine 1, the intake pressure PB (the absolute pressure in theintake pipe of the engine 1), and a detected value of the atmospherictemperature T_(A), from sensors (not shown) combined with the engine 1.The ROM of the control unit 16 corresponds to a recording mediumaccording to the present invention.

The control unit 16 has as its functional means an air-fuel ratiocontrol means 17 for controlling the air-fuel ratio in the exhaust gasemitted from the engine 1, and a sensor temperature control means 18 forcontrolling the temperature of the active element 10 of the O₂ sensor 8.

The air-fuel ratio control means 17 controls the air-fuel ratio in theexhaust gas supplied from the engine 1 to the catalytic converter 4 inorder to achieve a good purifying ability (purification rate) of thecatalytic converter 4 to purify CO (carbon monoxide), HC (hydrocarbon),and NOx (nitrogen oxide). When the O₂ sensor 8 of the above outputcharacteristics is disposed downstream of the catalytic converter 4, agood purifying ability of the catalytic converter 4 to purify CO, HC,and NOx can be achieved irrespective of the deteriorated state of thecatalytic converter 4 by controlling the air-fuel ratio in the exhaustgas supplied to the catalytic converter 4, i.e., the air-fuel ratio inthe exhaust gas upstream of the catalytic converter 4, to settle theoutput Vout of the O₂ sensor 8 at a certain predetermined value Vop (seeFIG. 3).

Specifically, the air-fuel ratio control means 17 uses the predeterminedvalue Vop as a target value for the output Vout of the O₂ sensor 8, andcontrols the air-fuel ratio in the exhaust gas supplied from the engine1 to the catalytic converter 4 in order to settle and keep the outputVout of the O₂ sensor 8 at the target value Vop. Such an air-fuel ratiocontrol process is carried out by determining a target air-fuel ratio inthe exhaust gas supplied to the catalytic converter 4 according to afeedback control process in order to converge the output Vout of the O₂sensor 8 to the target value Vop, and adjusting the amount of fuel to besupplied to the engine 1 according to a feedback control process inorder to converge the output KACT (a detected value of the air-fuelratio) of the wide-range air-fuel ratio sensor 9 to the target air-fuelratio. Specific details of the air-fuel ratio control process carried bythe air-fuel ratio control means 17 do not constitute an essentialfeature of the present invention, and will not be described below. Theair-fuel ratio control process carried by the air-fuel ratio controlmeans 17 is carried out as described in paragraphs [0071]-[0362] in thespecification of Japanese laid-open patent publication No. 11-324767 orU.S. Pat. No. 6,188,953, for example.

The output characteristics of the O₂ sensor 8 change depending on thetemperature of the active element 10 thereof. In FIG. 3, the solid-linecurve “a”, a broken-line curve “b”, a dot-and-dash-line curve “c”, and atwo-dot-and-dash-line curve “d” represent the output characteristics ofthe O₂ sensor 8 when the active element 10 has temperatures of 800° C.,750° C., 700° C., and 600° C., respectively. As can be seen from FIG. 3,if the temperature of the active element 10 changes in a temperaturerange lower than 750° C., then the gradient (sensitivity) of a change inthe output Vout of the O₂ sensor 8 in the vicinity of the stoichiometricair-fuel ratio (the high-sensitivity air-fuel ratio range Δ) and thelevel of the output Vout at air-fuel ratios richer than thehigh-sensitivity air-fuel ratio range Δ tend to change. If thetemperature of the active element 10 is 750° C. or higher, then a changein the output characteristics of the O₂ sensor 8 with respect to achange in the temperature of the active element 10 is so small that theoutput characteristics of the O₂ sensor 8 are substantially constant.

Since the output characteristics of the O₂ sensor 8 change depending onthe temperature of the active element 10 as described above, the controlproperties (stability and quick response) of the air-fuel ratio controlmeans 17 are likely to be lowered depending on the temperature of theactive element 10. This is because in controlling the air-fuel ratio inthe exhaust gas in order to keep the output Vout of the O₂ sensor 8 atthe target value Vop, the output characteristics of the O₂ sensor 8 inthe vicinity of the stoichiometric air-fuel ratio, i.e., the outputcharacteristics of the O₂ sensor 8 in the high-sensitivity air-fuelratio range Δ, are liable to greatly affect those control properties.The target value Vop for the output Vout of the O₂ sensor 8 to keep wellthe ability of the catalyst 7 of the catalytic converter 4 to purify theexhaust gas also changes depending on the temperature of the activeelement 10 in a temperature range lower than 750° C. Therefore, it ispreferable to keep the temperature of the active element 10 of the O₂sensor 8 basically at a constant level for the purpose of wellcontrolling the air-air ratio with the air-fuel ratio control means 17,i.e., controlling the output Vout of the O₂ sensor 8 at the target valueVop, and achieving a good purifying ability of the catalytic converter4.

If the temperature of the active element 10 of the O₂ sensor 8 is 750°C. or higher, then the output characteristics of the O₂ sensor 8 aresubstantially constant and stable. According to the inventors'knowledge, if the temperature of the active element 10 is kept at atemperature equal or higher than 750° C., e.g., 800° C., then the targetvalue Vop for the output Vout of the O₂ sensor 8 to keep well theability of the catalyst 7 of the catalytic converter 4 to purify theexhaust gas is present in an area denoted by Y on the curve “a” in FIG.3, i.e., an inflection point Y where the gradient of the curve “a”representing the output characteristics of the O₂ sensor 8 switches froma larger value to a smaller value as the air-fuel ratio becomes richer.At this time, the air-fuel ratio can be controlled to keep the outputVout of the O₂ sensor 8 at the target value Vop. The reason for theabove air-fuel fuel control appears to be that the sensitivity of theoutput Vout of the O₂ sensor 8 to the air-fuel ratio at the inflectionpoint Y is neither excessively high nor small, but is appropriate.

According to the present embodiment, the sensor temperature controlmeans 18 controls the ceramic heater 13 to keep the temperature of theactive element 10 of the O₂ sensor 8 at a desired temperature which isbasically equal to or higher than 750° C., e.g., 800° C. A controlprocess carried out by the sensor temperature control means 18 will bedescribed below.

As shown in FIG. 4, the sensor temperature control means 18 has as itsmajor functions an exhaust temperature observer 19 for sequentiallyestimating an exhaust gas temperature Tgd in the exhaust passage 3 nearthe O₂ sensor 8, i.e., at an intermediate portion of the exhaust pipe 6b, an element temperature observer 20 (temperature estimating means) forestimating the temperature T_(O2) of the active element 10 of the O₂sensor 8 and the temperature Tht of the ceramic heater 13 using theestimated value of the exhaust gas temperature Tgd, a target valuesetting means 21 for setting a target value R for the temperature of theactive element 10, and a heater controller 22 (heater control means) forcontrolling energization of the ceramic heater 13, i.e., controlling theelectric energy supplied to the ceramic heater 13, using the estimatedvalues of the temperature T_(O2) of the active element 10 and thetemperature Tht of the ceramic heater 13, the target value R, and theestimated value of the exhaust gas temperature Tgd.

In the present embodiment, the ceramic heater 13 is controlled for itsenergization (PWM control) by giving a pulsed voltage to a heaterenergization circuit (not shown). The amount of electric energy suppliedto the ceramic heater 13 is determined by the duty cycle DUT of thepulsed voltage (the ratio of the pulse duration to one period of thepulsed voltage). The heater controller 22 uses the duty cycle DUT of thepulsed voltage applied to the heater energization circuit as a controlinput (manipulated variable) for controlling the ceramic heater 13, andadjusts the duty cycle DUT to control the amount of electric energysupplied to the ceramic heater 13 and hence the amount of heat generatedby the ceramic heater 13. The duty cycle DUT generated by the heatercontroller 22 is also used in a processing sequence of the elementtemperature observer 20.

According to the present embodiment, the portion of the exhaust passage3 which extends from the exhaust port 2 of the engine 1 to the positionwhere the O₂ sensor 8 is located, i.e., the exhaust passage 3 upstreamof the O₂ sensor 8, is divided into a plurality of (four in the presentembodiment) partial exhaust passageways 3 a, 3 b, 3 c, 3 d along thedirection in which the exhaust passage 3 extends, i.e., the direction inwhich the exhaust gas flows. The exhaust temperature observer 19estimates, in a predetermined cycle time (period), the temperature ofthe exhaust gas at the exhaust port 2 (the inlet of the exhaust passage3) and the temperatures of the exhaust gas in the respective partialexhaust passageways 3 a, 3 b, 3 c, 3 d, or specifically, thetemperatures of the exhaust gas in the downstream ends of the respectivepartial exhaust passageways 3 a, 3 b, 3 c, 3 d, successively in thedownstream direction. Of the partial exhaust passageways 3 a, 3 b, 3 c,3 d, the partial exhaust passageways 3 a, 3 b are two partial exhaustpassageways divided from the exhaust passage 3 upstream of the catalyticconverter 4, i.e., the exhaust passage defined by the exhaust pipe 6 a,the partial exhaust passageway 3 c is a partial exhaust passagewayextending from the inlet to outlet of the catalytic converter 4, i.e.,the exhaust passage defined in the catalyst 7 in the catalytic converter4, and the partial exhaust passageway 3 d is a partial exhaustpassageway extending from the outlet of the catalytic converter 4 to theposition where the O₂ sensor 8 is located. The exhaust temperatureobserver 19 has its algorithm constructed as follows:

The temperature of the exhaust gas at the exhaust port 2 of the engine 1basically depends on the rotational speed NE and the intake pressure PBof the engine 1 while the engine 1 is operating in a steady state inwhich the rotational speed NE and the intake pressure PB are keptconstant. Therefore, the temperature of the exhaust gas at the exhaustport 2 can basically be estimated from detected values of the rotationalspeed NE and the intake pressure PB, which serve as parametersindicative of the operating state of the engine 1, based on apredetermined map which has been established by way of experimentation,for example. If the operating state (the rotational speed NE and theintake pressure PB) of the engine 1 varies, then the temperature of theexhaust gas at the exhaust port 2 suffers a time lag or delay in theresponse to the exhaust gas temperature determined by the map(hereinafter referred to as “basic exhaust gas temperatureTMAP(NE,PB)”).

According to the present embodiment, the exhaust temperature observer 19determines, in a predetermined cycle time (processing period), the basicexhaust gas temperature TMAP(NE,PB) from the detected values (latestdetected values) of the rotational speed NE and the intake pressure PBof the engine 1 based on the map, and thereafter sequentially estimatesan exhaust gas temperature Texg at the exhaust port 2 as a value whichfollows, with a time lag of first order, the basic exhaust gastemperature TMAP(NE,PB) as expressed by the following equation (1):Texg(k)=(1−Ktex)·(Texg(k−1 )+Ktex·TMAP(NE,PB)  (1)where k represents the ordinal number of a processing period of theexhaust temperature observer 19, and Ktex a coefficient (lagcoefficient) predetermined by way of experimentation or the like(0<Ktex<1). In the present embodiment, the intake pressure PB of theengine 1 serves as a parameter representative of the amount of intakeair introduced into the engine 1. Therefore, if a flow sensor is usedfor directly detecting the amount of intake air introduced into theengine 1, then the output of the flow sensor, i.e., a detected value ofthe amount of intake air, may be used instead of the detected value ofthe intake pressure PB.

Using the estimated value of the exhaust gas temperature Texg at theexhaust port 2, the temperatures of the exhaust gas in the respectivepartial exhaust passageways 3 a, 3 b, 3 c, 3 d are estimated asdescribed below. For illustrative purpose, a general heat transfer thatoccurs when a fluid flows through a circular tube 3 (see FIG. 5) whichextends in the direction of a Z-axis in the atmosphere while exchangingheat with the tube wall of the circular tube 3 will be described below.It is assumed that the fluid temperature Tg and the temperature Tw ofthe tube wall (hereinafter referred to as “circular tube temperatureTw”) are functions Tg(t,z), Tw(t,z) of the time t and the position z inthe direction of the Z-axis, the thermal conductivity of the tube wallof the circular tube 23 is infinite in the radial direction and nil inthe direction of the Z-axis. It is also assumed that the heat transferbetween the fluid and the tube wall of the circular tube 23 and the heattransfer between the tube wall of the circular tube 23 and the externalatmosphere are proportional to their temperature differences accordingto the Newton law of cooling. At this time, the following equations(2-1), (2-2) are satisfied:

$\begin{matrix}{{{{Sg} \cdot \rho}\;{g \cdot {Cg} \cdot \left( {\frac{\partial{Tg}}{\partial t} + {V \cdot \frac{\partial{Tg}}{\partial z}}} \right)}} = {\alpha\;{1 \cdot U \cdot \left( {{Tw} - {Tg}} \right)}}} & \left( {2\text{-}1} \right) \\{{{{Sw} \cdot \rho}\;{w \cdot {Cw} \cdot \frac{\partial{Tw}}{\partial t}}} = {{\alpha\;{1 \cdot U \cdot \left( {{Tg} - {Tw}} \right)}} + {\alpha\;{2 \cdot U \cdot \left( {T_{A} - {Tw}} \right)}}}} & \left( {2\text{-}2} \right)\end{matrix}$where Sg, ρg, Cg represent the density and specific heat of the fluidand the cross-sectional area of the fluid passage, respectively, Sw, ρw,Cw the density, specific heat, and cross-sectional area of the tube wallof the circular tube 23, respectively, V the speed of the fluid flowingthrough the circular tube 23, T_(A) the atmospheric temperature outsideof the circular tube 23, U the inner circumferential length of thecircular tube 23, α₁ the heat transfer coefficient between the fluid andthe tube wall of the circular tube 23, and α₂ the heat transfercoefficient between the tube wall of the circular tube 23 and theatmosphere. It is assumed that the atmospheric temperature T_(A) is keptconstant around the circular tube 23.

The above equations (2-1), (2-2) are modified into the followingequations (3-1), (3-2):

$\begin{matrix}{\frac{\partial{Tg}}{\partial t} = {{{- V} \cdot \frac{\partial{Tg}}{\partial z}} + {a \cdot \left( {{Tw} - {Tg}} \right)}}} & \left( {3\text{-}1} \right) \\{\frac{\partial{Tw}}{\partial t} = {{b \cdot \left( {{Tg} - {Tw}} \right)} + {c \cdot \left( {T_{A} - {Tw}} \right)}}} & \left( {3\text{-}2} \right)\end{matrix}$where a, b, c represent constants, α₁=·U/(Sg·ρg·Cg), b=α₁·U/(Sw·ρw·Cw),c=α₂·U/(Sw·ρw·Cw)).

The first term on the right side of the equation (3-1) is a shiftingflow term representing a time-dependent rate of change of the fluidtemperature Tg (a change in the temperature per unit time) depending onthe temperature gradient in the flowing direction of the fluid and thespeed of the fluid in a position z. The second term on the right side ofthe equation (3-1) is a heat transfer term representing a time-dependentrate of change of the fluid temperature Tg (a change in the temperatureper unit time) depending on the difference between the fluid temperatureTg and the circular tube temperature Tw in the position z, i.e., atime-dependent rate of change of the fluid temperature Tg which iscaused by the heat transfer between the fluid and the tube wall of thecircular tube 23. Therefore, the equation (3-1) indicates that thetime-dependent rate ∂Tg/∂t of change of the fluid temperature Tg in theposition z depends on the temperature change component of the shiftingflow term and the temperature change component of the heat transferterm, i.e., the sum of those temperature change components.

The first term on the right side of the equation (3-2) is a heattransfer term representing a time-dependent rate of change of thecircular tube temperature Tw (a change in the temperature per unit time)depending on the difference between the circular tube temperature Tw andthe fluid temperature Tg in the position z, i.e., a time-dependent rateof change of the circular tube temperature Tw which is caused by theheat transfer between the fluid and the tube wall of the circular tube23 in the position z. The second term on the right side of the equation(3-2) is a heat radiation term representing a time-dependent rate ofchange of the circular tube temperature Tw (a change in the temperatureper unit time) depending on the difference between the circular tubetemperature Tw and the atmospheric temperature T_(A) outside of thecircular tube 23 in the position z, i.e., a time-dependent rate ofchange of the circular tube temperature Tw depending on the heatradiation from the tube wall of the circular tube 23 into the atmospherein the position z. The equation (3-2) indicates that the time-dependentrate ∂Tw/∂t of change of the circular tube temperature Tw in theposition z depends on the temperature change component of the heattransfer term and the temperature change component of the heat radiationterm, i.e., the sum of those temperature change components.

According to the calculus of finite differences, the equations (3-1),(3-2) can be rewritten into the following equations (4-1), (4-2):

$\begin{matrix}{{{Tg}\left( {{t + {\Delta\; t}},z} \right)} = {{{Tg}\left( {t,z} \right)} - {\frac{{V \cdot \Delta}\; t}{\Delta\; z} \cdot \left( {{{Tg}\left( {t,z} \right)} - {{Tg}\left( {t,{z - {\Delta\; z}}} \right)}} \right)} + {{a \cdot \Delta}\;{t \cdot \left( {{{Tw}\left( {t,z} \right)} - {{Tg}\left( {t,z} \right)}} \right)}}}} & \left( {4\text{-}1} \right) \\{{{Tw}\left( {{t + {\Delta\; t}},z} \right)} = {{{Tw}\left( {t,z} \right)} + {{b \cdot \Delta}\;{t \cdot \left( {{{Tg}\left( {t,z} \right)} - {{Tw}\left( {t,z} \right)}} \right)}} + {{c \cdot \Delta}\;{t \cdot \left( {T_{A} - {{Tw}\left( {t,z} \right)}} \right)}}}} & \left( {4\text{-}2} \right)\end{matrix}$

The above equations (4-1), (4-2) indicate that if the fluid temperatureTg(t,z) and the circular tube temperature Tw(t,z) in the position z atthe time t, and the fluid temperature Tg(t,z−Δz) in a position z−Δzwhich precedes the position z (upstream thereof) at the time t areknown, then the fluid temperature Tg(t+Δt,z) and the circular tubetemperature Tw(t+Δt,z) in the position z at a next time t+Δt can bedetermined, and that the fluid temperatures Tg and the circular tubetemperatures Tw in successive positions z+Δz, z+2Δz, . . . can bedetermined by solving the equations (4-1), (4-2) simultaneously insequence for those positions. Specifically, if initial values of thefluid temperature Tg and the circular tube temperature Tw (initialvalues at t=0) are given in the positions z, z+Δz, z+2Δz, . . . and thefluid temperature Tg(t,0) at an origin (e.g., the inlet of the circulartube 23) in the direction of the Z-axis of the circular tube 23 is given(it is assumed that z−Δz=0), then the fluid temperatures Tg and thecircular tube temperatures Tw in successive positions z, z+Δz, z+2Δz, .. . . at successive times t, t+Δt, t+2Δt, . . . can be calculated.

The fluid temperature Tg(t,z) in the position z can be calculated bycumulatively adding (integrating), to the initial value Tg(0,z), thetemperature change component depending on the fluid speed V and thetemperature gradient in the position z (the temperature change componentrepresented by the second term of the equation (4-1)) and thetemperature change component depending on the difference between thefluid temperature Tg and the circular tube temperature Tw in theposition z (the temperature change component represented by the thirdterm of the equation (4-1)), at each given time interval. The fluidtemperatures in the other positions z+Δz, z+2Δz, . . . can similarly becalculated. The circular tube temperature Tw(t,z) in the position z canbe calculated by cumulatively adding (integrating), to the initial valueTw(0,z), the temperature change component depending on the differencebetween the fluid temperature Tg and the circular tube temperature Tw inthe position z (the temperature change component represented by thesecond term of the equation (4-2)) and the temperature change componentdepending on the difference between the circular tube temperature Tw andthe atmospheric temperature T_(A) in the position z (the temperaturechange component represented by the third term of the equation (4-2)),at each given time interval.

In the present embodiment, the exhaust temperature observer 19 uses themodel equations (4-1), (4-2) and determines the temperatures of theexhaust gas in the respective partial exhaust passageways 3 a, 3 b, 3 c,3 d as follows:

Of the partial exhaust passageways 3 a, 3 b, 3 c, 3 d, each of thepartial exhaust passageways 3 a, 3 b is defined by the exhaust pipe 6 a.In order to estimate the temperatures of the exhaust gas in the partialexhaust passageways 3 a, 3 b, the temperature changes depending on thespeed of the exhaust gas and the temperature gradient thereof (thetemperature gradient in the direction in which the exhaust gas flows),the heat transfer between the exhaust gas and the exhaust pipe 6 a, andthe heat radiation from the exhaust pipe 6 a into the atmosphere aretaken into account in the same manner as described above with respect tothe circular tube 23.

An estimated value of the exhaust gas temperature Tga in the partialexhaust passageway 3 a and an estimated value of the temperature Twa(hereinafter referred to as “exhaust pipe temperature Twa”) of theexhaust pipe 6 a in the partial exhaust passageway 3 a are determined byrespective model equations (5-1), (5-2), shown below, in each cycle timeof the processing sequence of the exhaust temperature observer 19. Anestimated value of the exhaust gas temperature Tgb in the partialexhaust passageway 3 b and an estimated value of the exhaust pipetemperature Twb in the partial exhaust passageway 3 b are determined byrespective model equations (6-1), (6-2), shown below, in each cycle timeof the processing sequence of the exhaust temperature observer 19. Morespecifically, the exhaust gas temperature Tga and the exhaust pipetemperature Twa that are determined by the equations (5-1), (5-2)represent estimated values of the temperatures in the vicinity of thedownstream end of the partial exhaust passageway 3 a. Likewise, theexhaust gas temperature Tgb and the exhaust pipe temperature Twb thatare determined by the equations (6-1), (6-2) represent estimated valuesof the temperatures in the vicinity of the downstream end of the partialexhaust passageway 3 b.

$\begin{matrix}{{{Tga}\left( {k + 1} \right)} = {{{Tga}(k)} - {{Vg} \cdot \frac{d\; t}{La} \cdot \left( {{{Tga}(k)} - {{Texg}(k)}} \right)} + {{{Aa} \cdot d}\;{t \cdot \left( {{{Twa}(k)} - {{Tga}(k)}} \right)}}}} & \left( {5\text{-}1} \right) \\{{{Twa}\left( {k + 1} \right)} = {{{Twa}(k)} + {{{Ba} \cdot d}\;{t \cdot \left( {{{Tga}(k)} - {{Twa}(k)}} \right)}} + {{{Ca} \cdot d}\;{t \cdot \left( {{T_{A}(k)} - {{Twa}(k)}} \right)}}}} & \left( {5\text{-}2} \right) \\{{{Tgb}\left( {k + 1} \right)} = {{{Tgb}(k)} - {{Vg} \cdot \frac{d\; t}{Lb} \cdot \left( {{{Tgb}(k)} - {{Tga}(k)}} \right)} + {{{Ab} \cdot d}\;{t \cdot \left( {{{Twb}(k)} - {{Tgb}(k)}} \right)}}}} & \left( {6\text{-}1} \right) \\{{{Twb}\left( {k + 1} \right)} = {{{Twb}(k)} + {{{Bb} \cdot d}\;{t \cdot \left( {{{Tgb}(k)} - {{Twb}(k)}} \right)}} + {{{Cb} \cdot d}\;{t \cdot \left( {{T_{A}(k)} - {{Twb}(k)}} \right)}}}} & \left( {6\text{-}2} \right)\end{matrix}$

In the equations (5-1), (5-2), (6-1), (6-2), dt represents the period(cycle time) of the processing sequence of the exhaust temperatureobserver 19, and corresponds to At in the equations (4-1), (4-2). In theequations (5-1), (6-1), La, Lb represent the respective lengths (fixedvalues) of the partial exhaust passageways 3 a, 3 b, and correspond toAz in the equation (4-1). Aa, Ba, Ca in the equations (5-1), (5-2) andAb, Bb, Cb in the equations (6-1), (6-2) represent model coefficientscorresponding respectively to a, b, c in the equations (4-1), (4-2), andthe values of those model coefficients are set (identified) in advanceby way of experimentation or simulation. In the equations (5-1), (6-1),Vg represents a parameter (to be determined as described later on)indicative of the speed of the exhaust gas, and corresponds to V in theequation (4-1).

The exhaust gas temperature Texg(k) (the exhaust gas temperature at theexhaust port 2) which is required to calculate a new estimated valueTga(k+1) of the exhaust gas temperature Tga according to the equation(5-1) is basically of the latest value determined according to theequation (1). Similarly, the exhaust gas temperature Tga(k) (the exhaustgas temperature in the partial exhaust passageway 3 a) which is requiredto calculate a new estimated value Tgb(k+1) of the exhaust gastemperature Tgb according to the equation (6-1) is basically of thelatest value determined according to the equation (5-1). The atmospherictemperature T_(A)(k) which is required in the calculation of theequations (5-2), (6-2) is of the latest value of the atmospherictemperature detected by an atmospheric temperature sensor (in thepresent embodiment, a sensor on the engine 1 is used for thisatmospheric temperature sensor), not shown. In the present embodiment,the gas speed parameter Vg which is required in the calculation of theequations (5-1), (6-1) is of a value which is calculated from latestdetected values of the rotational speed NE and the intake pressure PBaccording to the following equation (7):

$\begin{matrix}{{Vg} = {\frac{NE}{NEBASE} \cdot \frac{PB}{PBBASE}}} & (7)\end{matrix}$where NEBASE, PBBASE represent a predetermined rotational speed and apredetermined intake pressure, which are set to, for example, themaximum rotational speed of the engine 1 and 760 mmHg (≈101 kPa),respectively. The gas speed parameter Vg calculated according to theequation (7) is proportional to the speed of the exhaust gas, with Vg≦1.

In the present embodiment, initial values Tga(0), Twa(0), Tgb(0), Twb(0)of the estimated values for the exhaust gas temperature Tga, the exhaustpipe temperature Twa, the exhaust gas temperature Tgb, and the exhaustpipe temperature Twb are set to the atmospheric temperature which isdetected by the atmospheric temperature sensor (not shown) when theengine 1 has started to operate (upon an engine startup).

The partial exhaust passageway 3 c is defined by the catalyst 7 in thecatalytic converter 4. The catalyst 7 generates heat by itself due toits own exhaust gas purifying action (specifically, anoxidizing/reducing action), and the amount of heat (the amount of heatper unit time) generated by the catalyst 7 is substantially inproportion to the speed of the exhaust gas. This is because as the speedof the exhaust gas is higher, the exhaust gas components reacting withthe catalyst 7 per unit time increase.

According to the present embodiment, for estimating the exhaust gastemperature in the partial exhaust passageway 3 c with high accuracy,the generation of heat by the catalyst 7 in the catalytic converter 4 aswell as the temperature change depending on the speed and temperaturegradient of the exhaust gas, the heat transfer between the exhaust gasand the catalyst 7, and the heat radiation from the catalyst 7 into theatmosphere are taken into account.

An estimated value of the exhaust gas temperature Tgc in the partialexhaust passageway 3 c and an estimated value of the temperature Twc(hereinafter referred to as “catalyst temperature Twc”) of the catalyst7 which defines the partial exhaust passageway 3 c are determined byrespective model equations (8-1), (8-2), shown below, in each cycle timeof the processing sequence of the exhaust temperature observer 19. Morespecifically, the exhaust gas temperature Tgc and the catalysttemperature Twc that are determined by the equations (8-1), (8-2)represent estimated values of the temperatures in the vicinity of thedownstream end of the partial exhaust passageway 3 c, i.e., in thevicinity of the outlet of the catalytic converter 4.

$\begin{matrix}{{{Tgc}\left( {k + 1} \right)} = {{{Tgc}(k)} - {{Vg} \cdot \frac{d\; t}{Lc} \cdot \left( {{{Tgc}(k)} - {{Tgb}(k)}} \right)} + {{{Ac} \cdot d}\;{t \cdot \left( {{{Twc}(k)} - {{Tgc}(k)}} \right)}}}} & \left( {8\text{-}1} \right) \\{{{Twc}\left( {k + 1} \right)} = {{{Twc}(k)} + {{{Bc} \cdot d}\;{t \cdot \left( {{{Tgc}(k)} - {{Twc}(k)}} \right)}} + {{Cc} \cdot {dt} \cdot \left( {{T_{A}(k)} - {{Twc}(k)}} \right)} + {{Dc} \cdot {dt} \cdot {Vg}}}} & \left( {8\text{-}2} \right)\end{matrix}$

In the equation (8-1), Lc represents the length (fixed value) of thepartial exhaust passageway 3 c, and corresponds to Δz in the equation(4-1). Ac, Bc, Cc in the equations (8-1), (8-2) represent modelcoefficients corresponding respectively to a, b, c in the equations(4-1), (4-2), and the values of those model coefficients are set(identified) in advance by way of experimentation or simulation. Thefourth term on the right side of the equation (8-2) represents atemperature change component of the catalyst 7 in the catalyticconverter 4 due to the heating of the catalyst 7 by itself, i.e., thetemperature change per period of the processing sequence of the exhausttemperature observer 19, and is proportional to the gas speed parameterVg. As with Ac through Cc, Dc in the fourth term represents a modelcoefficient is set (identified) in advance by way of experimentation orsimulation. Therefore, the equation (8-2) corresponds to the combinationof the right side of the equation (4-2) with a temperature changecomponent due to the heating of a passage-defining member (the catalyst7).

dt, Vg in the equations (8-1), (8-2) have the same meanings and valuesas those in the equations (5-1) through (6-2). The value of T_(A) usedin the calculation of the equation (8-2) is identical to those used inthe equation (5-2), (6-2). In the present embodiment, the initial valuesTgc(0), Twc(0) of the exhaust gas temperature Tgc and the catalysttemperature Twc are equal to the detected value of the atmospherictemperature at the time the engine 1 has started to operate, as with theequations (5-1) through (6-2).

The partial exhaust passageway 3 d is defined by the exhaust pipe 6 bsimilar to the exhaust pipe 6 a which define the partial exhaustpassageways 3 a, 3 b. The exhaust gas temperature Tgd in the partialexhaust passageway 3 d and the exhaust pipe temperature Twa of theexhaust pipe 6 b, or more specifically the temperature at the downstreamend of the partial exhaust passageway 3 d, are determined respectivelyby the following equations (9-1), (9-2) which are similar to theequations (5-1) through (6-2):

$\begin{matrix}{{{Tgd}\left( {k + 1} \right)} = {{{Tgd}(k)} - {{Vg} \cdot \frac{\mathbb{d}t}{L\mathbb{d}} \cdot \left( {{{Tgd}(k)} - {{Tgc}(k)}} \right)} + {{Ad} \cdot {dt} \cdot \left( {{{Twd}(k)} - {{Tgd}(k)}} \right)}}} & \left( {9\text{-}1} \right) \\{{{Twd}\left( {k + 1} \right)} = {{{Twd}(k)} + {{Bd} \cdot {dt} \cdot \left( {{{Tgd}(k)} - {{Twd}(k)}} \right)} + {{Cd} \cdot {dt} \cdot \left( {{T_{A}(k)} - {{Twd}(k)}} \right)}}} & \left( {9\text{-}2} \right)\end{matrix}$

In the equation (9-1), Ld represents the length (fixed value) of thepartial exhaust passageway 3 d, and corresponds to Δz in the equation(4-1). Ad, Bd, Cd in the equations (9-1), (9-2) represent modelcoefficients corresponding respectively to a, b, c in the equations(4-1), (4-2), and the values of those model coefficients are set(identified) in advance by way of experimentation or simulation.

dt, Vg in the equations (9-1), (9-2) have the same meanings and valuesas those in the equations (5-1) through (6-2). The value of T_(A) usedin the calculation of the equation (9-2) is identical to those used inthe equation (5-2), (6-2), (8-2). The initial values Tgd(0), Twd(0) ofthe exhaust gas temperature Tgd and the catalyst temperature Twd areequal to the detected value of the atmospheric temperature at the timethe engine 1 has started to operate, as with the equations (5-1) through(6-2).

The processing sequence of the exhaust temperature observer 19, asdescribed above, determines estimated values of the exhaust gastemperatures Texg, Tga, Tgb, Tgc, Tgd in the exhaust port 2 of theengine 1 and the partial exhaust passageways 3 a, 3 b, 3 c, 3 dsuccessively downstream in each cycle time. The estimated value of theexhaust gas temperature Tgd in the partial exhaust passageway 3 d whichis located most downstream corresponds to the temperature of the exhaustgas in the vicinity of the location of the O₂ sensor 8. The estimatedvalue of the exhaust gas temperature Tgd is obtained as the estimatedvalue of the exhaust gas temperature in the vicinity of the location ofthe O₂ sensor 8.

The algorithm of the estimating process of the exhaust temperatureobserver 19 is shown in block form in FIG. 6. In FIG. 6, the modelequation (1) is referred to as an exhaust port thermal model 24, themodel equations (5-1), (5-2) and the model equations (6-1), (6-2) aspre-CAT exhaust system thermal models 25, 26, respectively, the modelequations (8-1), (8-2) as an in-CAT exhaust system thermal model 27, andthe model equations (9-1), (9-2) as a post-CAT exhaust system thermalmodel 28. As shown in FIG. 6, each of the thermal models 24 through 28is supplied with the detected values of the rotational speed NE and theintake pressure PB of the engine 1. The detected values of therotational speed NE and the intake pressure PB which are supplied to theexhaust port thermal model 24 are used to determine the basic exhaustgas temperature TMAP, and the detected values of the rotational speed NEand the intake pressure PB which are supplied to the exhaust systemthermal models 25 through 28 are used to determine the value of the gasspeed parameter Vg. Each of the thermal models 24 through 28 is alsosupplied with the detected value of the atmospheric temperature T_(A).The pre-CAT exhaust system thermal model 25, the pre-CAT exhaust systemthermal model 26, the in-CAT exhaust system thermal model 27, and thepost-CAT exhaust system thermal model 28 are supplied with the estimatedvalues of the exhaust gas temperatures Texg, Tga, Tgb, Tgc,respectively, which are outputted from the higher-level thermal models24, 25, 26, 27. The post-CAT exhaust system thermal model 28 eventuallyproduces the estimated value of the exhaust gas temperature Tgd in thevicinity of the location of the O₂ sensor 8.

In the present embodiment, the detected value produced by theatmospheric temperature sensor on the engine 1 is used to estimate thetemperatures of the passage-defining members (the exhaust pipe 6 a, thecatalyst 7 in the catalytic converter 4, and the exhaust pipe 6 b) whichdefine the partial exhaust passageways 3 a, 3 b, 3 c, 3 d. However, thedetected value produced by an atmospheric sensor which is disposedoutside of the exhaust passage 3 may be used to estimate thetemperatures of those passage-defining members.

The element temperature observer 20 will be described below. The elementtemperature observer 20 estimates the temperature T_(O2) of the activeelement 10 of the O₂ sensor 8 sequentially in given cycle times in viewof the thermal transfer between the active element 10 and the exhaustgas held in contact therewith and the thermal transfer between theactive element 10 and the ceramic heater 13 (hereinafter referred tosimply as “heater 13”) which heats the active element 10. The elementtemperature observer 20 also estimates the temperature Tht of the heater13 in order to estimate the temperature T_(O2) of the active element 10.In estimating the temperature Tht of the heater 13, the elementtemperature observer 20 takes into account the heat transfer between theheater 13 and the active element 13 and also the heating of the heater13 based on the electric energy supplied to the heater 13. The elementtemperature observer 20 has an estimating algorithm for estimating thetemperature T_(O2) and the temperature Tht, which is constructed asfollows:

The element temperature observer 20 determines an estimated value of thetemperature T_(O2) of the active element 10 (hereinafter referred to as“element temperature T_(O2)”) and an estimated value of the temperatureTht of the heater 13 (hereinafter referred to as “heater temperatureTht”) sequentially in given cycle times respectively according to themodel equations (10-1), (10-2) described below. The equation (10-1) isthe equation of an element temperature model, and the equation (10-2) isthe equation of a heater temperature model.T _(O2)(k+1)=T _(O2)(k)+Ax·dt·(Tgd(k)−T _(O2)(k))+Bx·dt·(Tht(k)−T_(O2)(k))  (10-1)Tht(k+1)=Tht(k)−Cx·dt·(Tht(k)−T _(O2)(k))+Dx·dt·DUT(k)  (10-2)

The equation (10-1) indicates that the temperature change of the activeelement 10 in each cycle time depends on a temperature change component(the second term on the right side of the equation (10-1)) depending onthe difference between the exhaust gas temperature Tgd in the vicinityof the location of the O₂ sensor 8 (the exhaust gas temperature in thepartial exhaust passageway 3 d) and the element temperature T_(O2),i.e., a temperature change component which is caused by the heattransfer between the active element 10 and the exhaust gas held incontact therewith, and a temperature change component (the third term onthe right hand of the equation-(10-1)) depending on the differencebetween the element temperature T_(O2) and the heater temperature Tht,i.e., a temperature change component which is caused by the heattransfer between the active element 10 and the ceramic heater 13, i.e.,the sum of those temperature change components.

The equation (10-2) indicates that the temperature change of the heater13 in each cycle time depends on a temperature change component (thesecond term on the right side of the equation (10-2)) depending on thedifference between the element temperature T_(O2) and the heatertemperature Tht, i.e., a temperature change component which is caused bythe heat transfer between the active element 10 and the heater 13, and atemperature change component depending on the duty cycle DUT that isgenerated by the heat controller 22 as described later on, i.e., atemperature change component which is caused by the heating of theheater 13 based on the electric energy supplied thereto, i.e., the sumof those temperature change components.

In the equations (10-1), (10-2), Ax, Bx, Cx, Dx represent modelcoefficients whose values are set (identified) in advance by way ofexperimentation or simulation, and dt represents the period (cycle time)of the processing sequence of the element temperature observer 20. Inthe present embodiment, the period dt is set to the same value as thecycle time (represented by dt in the equations (5-1) through (9-2)) ofthe processing sequence of the exhaust temperature observer 19.

The duty cycle DUT(k) which is required in the calculation of theequation (10-2) is of the latest value of the duty cycle DUT that iscalculated by the heater controller 22 as described later on. In thepresent embodiment, the initial values T_(O2)(0), Tht(0) of the elementtemperature TO2 and the heater temperature Tht are equal to the detectedvalue of the atmospheric temperature at the time the engine 1 hasstarted to operate.

The element temperature observer 20 sequentially calculates theestimated values of the element temperature T_(O2) and the heatertemperature Tht according to the estimating algorithm described above.

The heater controller 22 will be described below. The heater controller22 sequentially generates the duty cycle DUT as a control input(manipulated variable) for controlling the heater 13 according to anoptimum predictive control algorithm, and controls the electric energysupplied to the heater 13 with the generated duty cycle DUT.

According to the present embodiment, attention is paid to the differencebetween the element temperature T_(O2) and a target value therefor, achange per given time in the difference (corresponding to a rate ofchange of the difference), and a change per given time in the heatertemperature Tht (corresponding to a rate of change of the heatertemperature Tht), and model equations for an object to be controlled bythe heater controller 22 are introduced using the above differences andchanges as state quantities relative to the object to be controlled bythe heater controller 22. The heater controller 22 has its algorithmconstructed as described below.

First, model equations for the object to be controlled by the heatercontroller 22 will be described below. Changes ΔT_(O2), ΔTht per giventime in the element temperature T_(O2) and the heater temperature Thtare expressed by the following equations (11-1), (11-2) based on therespective model equations (10-1), (10-2) with respect to the elementtemperature observer 20:

$\begin{matrix}{{\Delta\;{T_{O2}\left( {k + 1} \right)}} = {{\Delta\;{T_{O2}(k)}} + {A\;{x \cdot {dt} \cdot \left( {{\Delta\;{{Tgd}(k)}} - {\Delta\;{T_{O2}(k)}}} \right)}} +}} & \left( {11\text{-}1} \right) \\{\mspace{160mu}{{Bx} \cdot {dt} \cdot \left( {{\Delta\;{{Tht}(k)}} - {\Delta\;{T_{O2}(k)}}} \right)}} & \; \\{\mspace{124mu}{= {{{\left( {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}}} \right) \cdot \Delta}\;{T_{O2}(k)}} +}}} & \; \\{\mspace{160mu}{{{{Ax} \cdot {dt} \cdot \Delta}\;{{Tgd}(k)}} + {{{Bx} \cdot {dt} \cdot \Delta}\;{{Tht}(k)}}}} & \; \\{{\Delta\;{{Tht}\left( {k + 1} \right)}} = {{\Delta\;{{Tht}(k)}} - {{Cx} \cdot {dt} \cdot \left( {{\Delta\;{{Tht}(k)}} - {\Delta\;{T_{O2}(k)}}} \right)} +}} & \left( {11\text{-}2} \right) \\{\mspace{155mu}{{{Dx} \cdot {dt} \cdot \Delta}\;{{DUT}(k)}}} & \; \\{\mspace{124mu}{= {{{\left( {1 - {{Cx} \cdot {dt}}} \right) \cdot \Delta}\;{{Tht}(k)}} + {{{Cx} \cdot {dt} \cdot \Delta}\;{T_{O2}(k)}} +}}} & \; \\{\mspace{155mu}{{{Dx} \cdot {dt} \cdot \Delta}\;{{DUT}(k)}}} & \;\end{matrix}$

In the above equations (11-1), (11-2), ΔT_(O2)(k)=T_(O2)(k+1)−T_(O2)(k),ΔTht(k)=Tht(k+1)−Tht(k), ΔTgd(k)=Tgd(k+1)−Tgd(k),ADUT(k)=DUT(k+1)−DUT(k).

A target value for the element temperature T_(O2) is represented by R,and the difference e between the element temperature T_(O2) and thetarget value R, i.e., the difference in each cycle time (hereinafterreferred to as “element temperature difference e”), is defined accordingto the following equation (12):e(k)=T _(O2)(k)−R(k)  (12)

A change Δe in the element temperature difference e in each cycle time(hereinafter referred to as “element temperature difference change Δe”)is expressed by the following equation (13) based on the above equations(11-1), (12):

$\begin{matrix}\begin{matrix}{{\Delta\;{e\left( {k + 1} \right)}} = {{\Delta\;{T_{O2}\left( {k + 1} \right)}} - {\Delta\;{R\left( {k + 1} \right)}}}} \\{= {{{\left( {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}}} \right) \cdot \Delta}\;{e(k)}} + {{{Ax} \cdot {dt} \cdot \Delta}\;{{Tgd}(k)}} +}} \\{{{{Bx} \cdot {dt} \cdot \Delta}\;{{Tht}(k)}} - {\Delta\;{R\left( {k + 1} \right)}} +} \\{{\left( {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}}} \right) \cdot \Delta}\;{R(k)}}\end{matrix} & (13)\end{matrix}$

In the equation (13), Δe(k)=e(k+1)−e(k), ΔR(k)=R(k+1)−R(k). In derivingthe equation (13), the equation ΔT_(O2)=Δe(k)+ΔR(k) (based on theequation (12)) is employed.

The equation ΔT_(O2)=Δe(k)+ΔR(k) is applied to the equation (11-2), andthe resulting equation is modified into the following equation (14):ΔTht(k+1)=(1−Cx·dt)·ΔTht(k)+Cx·dt·Δe(k)+Dx·dt·ΔDUT(k)+Cx·dt·AR(k)  (14)

If a state quantity vector X0(k)=(e(k),Δe(k), ΔTht(k))^(T) (T representsa transposition) is introduced, then the following equation (15) isobtained from the equations (14), (15) and the equatione(k+1)=e(k)+Δe(k):X0(k+1)=Φ·X0(k)+G·ΔDUT(k)+Gd·ΔTgd(k)+Gr·R0(k+1)  (15)where

-   X0(k)=(e(k),Δe(k),ΔTht(k))^(T),-   R0(k+1)=(ΔR(k+1),ΔR(k))^(T),-   G=(0,0,Dx·dt)^(T),-   Gd=(0,Ax·dt,0)^(T),

$\begin{matrix}{\Phi = \begin{bmatrix}1 & 1 & 0 \\0 & {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}}} & {{Bx} \cdot {dt}} \\0 & {{Cx} \cdot {dt}} & {1 - {{Cx} \cdot {dt}}}\end{bmatrix}} \\{{Gr} = \begin{bmatrix}0 & 0 \\{- 1} & {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}}} \\0 & {{Cx} \cdot {dt}}\end{bmatrix}}\end{matrix}$In the equation (15), R0, G, Gd represent vectors defined in the abovedefinition clause, and Φ, Gr represent matrixes defined in the abovedefinition clause.

The above equation (15) is a basic equation of the model of the objectto be controlled by the heater controller 22.

In the above description, the period of the control process of theheater controller 22 is the same as the period dt of the processingsequences of the exhaust temperature observer 19 and the elementtemperature observer 20. Therefore, the period dt is used in the vectorsG, Gd and the matrixes Φ, Gr in the equation (15). It is preferable tocarry out the processing sequences of the exhaust temperature observer19 and the element temperature observer 20 in a relatively short period(e.g., a period of 20 through 50 msec.) in order to increase theaccuracy with which to estimate the temperatures. However, the period ofthe control process of the heater controller 22 may be longer than theperiod dt of the processing sequences of the exhaust temperatureobserver 19 and the element temperature observer 20 because the responsespeed of a change in the element temperature in response to the controlinput (duty cycle DUT) is relatively low (several Hz in terms offrequencies). According to an optimum predictive control process to bedescribed later on, since future values of the target value R of theelement temperature T_(O2) need to be stored and held for a certaintime, the storage capacity of a memory for storing the target value Rbecomes large if the period of the control process of the heatercontroller 22 is short.

According to the present embodiment, the period (cycle time) of thecontrol process of the heater controller 22 is set to a value dtc (e.g.,300 through 500 msec.) longer than the period dt of the processingsequences of the exhaust temperature observer 19 and the elementtemperature observer 20.

In the present embodiment, the model equation of the object to becontrolled by the heater controller 22 is rewritten from the equation(15) into the following equation (16), using the period dtc of thecontrol process of the heater controller 22:X0(n+1)=Φ·X0(n)+G·ΔDUT(n)+Gd·ΔTgd(n)+Gr·R0(n+1)  (16)where

-   X0(n)=(e(n),Δe(n),ΔTht(n))^(T),-   R0(n+1)=(ΔR(n+1),ΔR(n))^(T),-   G=(0,0,Dx·dtc)^(T),-   Gd=(0,Ax·dtc,0)^(T),

$\Phi = \begin{bmatrix}1 & 1 & 0 \\0 & {1 - {{Ax} \cdot {dtc}} - {{Bx} \cdot {dtc}}} & {{Bx} \cdot {dtc}} \\0 & {{Cx} \cdot {dtc}} & {1 - {{Cx} \cdot {dtc}}}\end{bmatrix}$ ${Gr} = \begin{bmatrix}0 & 0 \\{- 1} & {1 - {{Ax} \cdot {dtc}} - {{Bx} \cdot {dtc}}} \\0 & {{Cx} \cdot {dtc}}\end{bmatrix}$

The equation (16) is a model equation of the object to be controlledwhich is actually used in the algorithm of the control process of theheater controller 22. In the equation (16), n represents the ordinalnumber of the period dtc of the control process of the heater controller22.

Using the above model equation, the algorithm of the control process ofthe heater controller 22, i.e., the algorithm of the optimum predictivecontrol process, is constructed as follows: It is assumed that thetarget value R of the element temperature T_(O2) is set for the futureuntil after Mr steps (until after a multiple by Mr of the period dtc ofthe control process of the heater controller 22), and the exhaust gastemperature Tgd which acts as a disturbance input is known in the futureuntil after Md steps (until after a multiple by Md of the period dtc ofthe control process of the heater controller 22). The value Mr will bereferred to as a target value predicting time Mr, and the value Md as anexhaust gas temperature predicting time Md. These predicting times Mr,Md are represented by integers whose unit is one period dtc of thecontrol process of the heater controller 22.

A controller for generating a control input ΔDUT for minimizing thevalue of an evaluating function J0 according to the following equation(17) serves as an optimum predictive servo controller:

$\begin{matrix}{{J0} = {\sum\limits_{n = {M + 1}}^{\infty}\left\lbrack {{{{X0}^{T}(n)} \cdot {Q0} \cdot {{X0}(n)}} + {\Delta\;{{{DUT}^{T}(n)} \cdot {H0} \cdot \Delta}\;{{DUT}(n)}}} \right\rbrack}} & (17)\end{matrix}$where M represents a larger one of the target value predicting time Mrand the exhaust gas temperature predicting time Md, i.e., M=max(Mr,Md),and Q0, H0 are weighted matrixes for adjusting the convergence of thestate quantity vector X0 and the power (size) of the control input ADUT.Q0 represents a 3-row, 3-column diagonal matrix as X0 is a cubic matrix,and H0 is a Scalar quantity as ΔDUT is a Scalar quantity. In the presentembodiment, in order to reduce the power consumption of the heater 13,Q0 is set to a unit matrix (a diagonal matrix whose all diagonalcomponents are “1”) and H0 is set to a value (e.g., 1000) greater thanthe diagonal components of the matrix Q0. The target value predictingtime Mr is set to 20, for example, and the exhaust gas temperaturepredicting time Md is set to 10, for example, with the period of thecontrol process of the heater controller 22 being in the range from 300to 500 msec.

The control input ΔDUT for minimizing the value of the evaluatingfunction according to the equation (17) is expressed by the equation(18) given below. In the present embodiment, it is assumed that theexhaust gas temperature Tgd is maintained at the present value in thefuture until after Md steps.

$\begin{matrix}{{\Delta\;{{DUT}(n)}} = {{{{F0} \cdot {X0}}(n)} + {\sum\limits_{i = 1}^{Mr}\left\lbrack {{{Fr0}(i)} \cdot {{R0}\left( {n + i} \right)}} \right\rbrack} + {{{Fdt} \cdot \Delta}\;{{Tgd}(n)}}}} & (18)\end{matrix}$

In the equation (18), F0 in the first term on the right side representsa cubic row vector (Fs0,Fe0,Fx0), Fr0(i) (i=1, 2, . . . , Mr) in thesecond term (the term of Σ) on the right side represent quadratic rowvectors (Fr01(i), Fr02(i)), and Fdt in the third term on the right siderepresents a Scalar quantity. They are expressed by the equations (19-1)through (19-3) given below. In the present embodiment, since it isassumed that the exhaust gas temperature Tgd is maintained at thepresent value in the future until after Md steps, Fdt in the third termof the right side represents a Scalar quantity. If Tgd in each step inthe future can be detected or estimated, then the control input DUT canbe determined using those Tgd. In such a case, Fdt represents a vectorcomprising elements (Md+1 elements) in {} of the equation (19-3).

$\begin{matrix}{{{F0} \equiv \left( {{Fs0},{Fe0},{Fx0}} \right)}\mspace{31mu} = {{- \left\lbrack {{H0} + {G^{T} \cdot P \cdot G}} \right\rbrack^{- 1}} \cdot G^{T} \cdot P \cdot \Phi}} & \left( {19\text{-}1} \right) \\{{{{Fr0}(i)} \equiv {\left( {{{Fr01}(i)},{{Fr02}(i)}} \right)\left( {{i = 1},2,\cdots\mspace{14mu},{Mr}} \right)}}\mspace{65mu} = {{- \left\lbrack {{H0} + {G^{T} \cdot P \cdot G}} \right\rbrack^{- 1}} \cdot G^{T} \cdot \left( \zeta^{T} \right)^{i - 1} \cdot P \cdot {Gr}}} & \left( {19\text{-}2} \right) \\{{Fdt} = {\sum\limits_{i = 0}^{Md}\left\{ {{- \left( {{H\; 0} + {G^{T} \cdot P \cdot G}} \right\rbrack^{- 1}} \cdot G^{T} \cdot \left( \zeta^{T} \right)^{i} \cdot P \cdot {Gd}} \right\}}} & \left( {19\text{-}3} \right)\end{matrix}$where P represents a matrix (a 3-row, 3-column matrix) satisfying thefollowing Ricatti equation (20-1), and ζ represents a matrix (a 3-row,3-column matrix) expressed by the following equation (20-2):P=Q0+Φ^(T) ·P·Φ−Φ·P·G [H0+G ^(T) ·P·G] ⁻¹ ·G ^(T) ·P·Φ  (20-1)ζ=Φ+G·F0  (20-2)

G, Gr, Gd, and Φ in the equations (19-2) through (19-3) and theequations (20-1), (20-2) are defined in the definition clause for theequation (16), and H0, Q0 in those equations represent weighted matrixesof the evaluating function J0 according to the equation (17) (H0 is aScalar quantity).

The second term (the term of Σ) on the right side of the equation (18)is rewritten using the components of Fr0, R0 (see the definition clausesfor the equations (19-2), (16)), and then modified into the followingequation (21):

$\begin{matrix}{{{\sum\limits_{i = 1}^{Mr}\left\lbrack {{{Fr0}(i)} \cdot {{R0}\left( {n + i} \right)}} \right\rbrack} = {\sum\limits_{i = 1}^{Mr}\left\lbrack {{{{Fr}(i)} \cdot \Delta}\;{R\left( {n + 1} \right)}} \right\rbrack}}{where}{{{Fr}(i)} = \left\lbrack \begin{matrix}{{Fr02}(1)} & {{:i} = 0} \\{{{Fr01}(i)} + {{Fr02}\left( {i + 1} \right)}} & {{{:i} = 1},2,\cdots\mspace{14mu},{{Mr} - 1}} \\{{Fr01}({Mr})} & {{:i} = {Mr}}\end{matrix} \right.}} & (21)\end{matrix}$

By putting the equation (21) into the equation (18) and rewriting thefirst term on the right side of the equation (18) using the componentsof F0, X0 (see the definition clauses for the equations (19-1), (16)),the equation (18) is expressed by the following equation (22):

$\begin{matrix}{{\Delta\;{{DUT}(n)}} = {{{Fs0} \cdot {e(n)}} + {{{Fe0} \cdot \Delta}\;{e(n)}} + {{{Fx0} \cdot \Delta}\;{{Tht}(n)}} + {\sum\limits_{i = 0}^{Mr}\left\lbrack {{{{Fr}(i)} \cdot \Delta}\;{R\left( {n + i} \right)}} \right\rbrack} + {{{Fdt} \cdot \Delta}\;{{Tgd}(n)}}}} & (22)\end{matrix}$

Since the control input DUT(n) to be generated by the heater controller22 is represented by the sum of its initial value DUT(0) and ΔDUT(1),ΔDUT(2), . . . , ΔDUT(n) cumulatively added thereto, the followingequation (23) is obtained from the above equation (22):

$\begin{matrix}{{{DUT}(n)} = {{{Fs0} \cdot {\sum\limits_{j = 1}^{n}{e(j)}}} + {{Fe0} \cdot {e(n)}} + {{Fx0} \cdot {{Tht}(n)}} + {\sum\limits_{i = 0}^{Mr}\left\lbrack {{{Fr}(i)} \cdot {R\left( {n + i} \right)}} \right\rbrack} + {{Fdt} \cdot {{Tgd}(n)}} - {{Fe0} \cdot {e(0)}} - {{Fx0} \cdot {{Tht}(0)}} - {\sum\limits_{i = 0}^{Mr}\left\lbrack {{{Fr}(i)} \cdot {R\left( {0 + i} \right)}} \right\rbrack} - {{Fdt} \cdot {{Tgd}(0)}} + {{DUT}(0)}}} & (23)\end{matrix}$

By setting the initial value terms of the equation (23), i.e., the sixthterm (the term of Fe0·e(0)) through the tenth term (DUT(0)), to “0”, thefollowing equation (24) is obtained as an equation for calculating thecontrol input DUT(n) to be actually generated by the heater controller22:

$\begin{matrix}{{{DUT}(n)} = {{{Fs0} \cdot {\sum\limits_{j = 1}^{n}{e(j)}}} + {{Fe0} \cdot {e(n)}} + {{Fx0} \cdot {{Tht}(n)}} + {\sum\limits_{i = 0}^{Mr}\left\lbrack {{{Fr}(i)} \cdot {R\left( {n + i} \right)}} \right\rbrack} + {{Fdt} \cdot {{Tgd}(n)}}}} & (24)\end{matrix}$

The equation (24) is a formula for calculating the control input DUT(n)(duty cycle) for controlling the heater 13 with the heater controller22. Specifically, the heater controller 22 sequentially calculates thecontrol input DUT(n) according to the equation (24) in each cycle time(period) of the control process of the heater controller 22, and appliesa pulsed voltage having the duty cycle DUT(n) to the heater energizationcircuit (not shown) to adjust the electric energy supplied to the heater13. The first through third terms (the term including Σe(j) through theterm including Tht(n)) of the equation (24) represent a control inputcomponent (a feedback component which will hereinafter be referred to as“optimum F/B component Uopfb”) depending on the element temperaturedifference e and the heater temperature Tht. The fourth term (the termof ΣFr(i)·R(n+1)) on the right side of the equation (24) represents acontrol input component (a feed-forward component which will hereinafterbe referred to as “optimum target value F/F component Uopfr”) dependingon the target value. The fifth term (the term including Tgd(n))represents a control input component (a feed-forward component whichwill hereinafter be referred to as “optimum disturbance F/F componentUopfd”) depending on the exhaust gas temperature Tgd (which functions asa disturbance on the object to be controlled). The heater controller 22which determines DUT as a control input according to the equation (24)is expressed in block form as shown in FIG. 7.

Fs0, Fe0, Fx0 which are required to determine the control input DUT(n)according to the equation (24) are of values calculated in advanceaccording to the equation (19-1). Fr(i) (i=0, 1, . . . , Mr) is ofvalues calculated in advance according to the equations (21), (19-2).Fdt is of a value calculated in advance according to the equation(19-3). These coefficients Fs0, Fe0, Fx0, Fr(i), Fdt may not necessarilybe of the values according to the defining equations, but may be ofvalues adjusted by way of simulation or experimentation. Furthermore,the coefficients Fs0, Fe0, Fx0, Fr(i), Fdt may be changed depending onthe element temperature, the heater temperature, etc.

The heater temperature Tht and the exhaust gas temperature Tgd which arerequired in the calculation of the equation (24) are of the latestestimated value of the heater temperature Tht determined by the elementtemperature observer 20 and the latest estimated value of the exhaustgas temperature Tgd determined by the exhaust temperature observer 19.

The element temperature difference e required in the calculation of theequation (24) is calculated from the latest estimated value of theelement temperature T_(O2) determined by the element temperatureobserver 20 and the target value R which has been set in a cycle timeprior to the target value predicting time Mr by the target value settingmeans 21.

The target value setting means 21 basically sets a temperature (e.g.,800° C. in the present embodiment) equal to or higher than 750° C. atwhich the output characteristics of the O₂ sensor 8 are stably good, asthe target value R for the temperature of the active element 10 in thesame cycle time as the cycle time (period) of the processing sequence ofthe heater controller 22. In order to perform the processing sequence ofthe heater controller 22 according to the algorithm of the optimumpredictive control process, the target value setting means 21 sets thetarget value R in each cycle time as a target value R(n+Mr) after thetarget value predicting time Mr from the present cycle time, and storesa series of target values R(n+Mr) for the target value predicting timeMr. Specifically, the target value setting means 21 stores Mr+1 targetvalues R(n), R(n+1), . . . , R(n+Mr) while sequentially updating them.The target value R used to determine the element temperature differencee that is required in the calculation of the equation (24) is the valueR(n) set and stored by the target value setting means 21 as describedabove in the cycle time prior to the target value predicting time Mr.The target values R(n), R(n+1), . . . , R(n+Mr) stored as describedabove are used to determine the value of the fourth term (the term of Σincluding R(n+i)) of the equation (24).

If the target value R of the element temperature T_(O2) is set to a hightemperature such as 800° C. from the start of operation of the engine 1,then the active element 10 tends to be damaged due to stresses caused byquick heating if water is applied to the active element 10 of the O₂sensor 8 when the engine 1 starts to operate. In the present invention,therefore, until a certain time (e.g., 15 seconds) elapses from thestart of operation of the engine 1, the target value setting means 21sets the target value R of the element temperature T_(O2) to atemperature lower than 750° C., e.g., 600° C.

Overall operation of the apparatus, particularly, the sensor temperaturecontrol means 18, according to the present embodiment will be describedbelow.

When the engine 1 starts to operate, the sensor temperature controlmeans 18 executes a main routine shown in FIG. 8 in a predeterminedcycle time. The period in which the main routine is executed is shorterthan the period dt of the processing sequence of the element temperatureobserver 20 and hence shorter than the period dtc of the processingsequence of the target value setting means 21 and the heater controller22.

The sensor temperature control means 18 acquires detected values of therotational speed NE and the intake pressure PB of the engine 1 and theatmospheric temperature T_(A) in STEP1, and then determines the value ofa countdown timer COPC for measuring the time dtc of one period of theprocessing sequence of the target value setting means 21 and the heatercontroller 22 in STEP2. The value of the countdown timer COPC has beeninitialized to “0” at the time when the engine 1 starts to operate.

If COPC=0, then the sensor temperature control means 18 newly sets thevalue of the countdown timer COPC to a timer setting time TM1 whichcorresponds to the period dtc of the control processes of the targetvalue setting means 21 and the heater controller 22 in STEP3.Thereafter, the target value setting means 21 carries out a process ofsetting a target value R for the element temperature T_(O2) of the O₂sensor 8 in STEP4, and the heater controller 22 carries out a process ofcalculating a duty cycle DUT of the heater 13 in STEP5. If COPC≠0 inSTEP2, then the sensor temperature control means 18 counts down thevalue of the countdown timer COPC in STEP6, and skips the processing inSTEP4 and STEP5. Therefore, the processing in STEP4 and STEP5 is carriedout at the period dtc determined by the timer setting time TM1.

The processing in STEP4 and STEP5 is specifically carried out asfollows: First, the processing in STEP4 is carried out by the targetvalue setting means 21 as shown in FIG. 9.

The target value setting means 21 compares the value of a parameter TSHrepresentative of the time that has elapsed from the start of the engine1 with a predetermined value XTM in STEP4-1. If TSH≦XTM, i.e., if theengine 1 is in a state immediately after it has started to operate, thenthe target value setting means 21 sets the target value R for theelement temperature T_(O2) to a low temperature (e.g., 600° C.) in orderto prevent damage to the active element 10 of the O₂ sensor 8 inSTEP4-2. Specifically, the target value R that is set at this time is atarget value R(n+Mr) after the target value predicting time Mr from thepresent.

If TSH>XTM in STEP4-1, then the target value setting means 21 sets thetarget value R for the element temperature T_(O2) from the presentdetected value (acquired in STEP1 shown in FIG. 8) of the atmospherictemperature TA based on a predetermined table in STEP4-3. The targetvalue R that is set at this time is basically a predetermined value(800° C. in the present embodiment) equal to or higher than 750° C. ifthe atmospheric temperature T_(A) is a normal temperature (e.g.,T_(A)>0° C.). When the atmospheric temperature T_(A) is low (e.g.,T_(A)<0° C.) as when the engine 1 is operating in a cold climate, if thetarget value R for the element temperature T_(O2) is a high temperatureof 800° C., the temperature of the heater 13 is liable to be excessivelyhigh. In the present embodiment, when the temperature of the heater 13becomes excessively high, the heater 13 is forcibly de-energized by anoverheating prevention process (described later on) to prevent itselffrom a failure. In STEP4-3, according to the present embodiment, whenthe atmospheric temperature T_(A) is low (e.g., T_(A)<0° C.), the targetvalue R for the element temperature T_(O2) is set to a value slightlylower than the normal value (e.g., 750° C.≦R 800° C.).

Specifically, as with the target value R set in STEP4-2, the targetvalue R set in STEP4-3 is a target value R(n+Mr) after the target valuepredicting time Mr from the present.

After having set the target value R (=R(n+Mr)) in STEP4-2 or STEP4-3,the target value setting means 21 updates the values of Mr+1 buffersRBF(0), RBF(1), . . . , RBF(Mr) for storing target values R for thetarget value predicting time Mr in STEP4-4, STEP4-5. The processing inSTEP4 is now finished.

In STEP4-4, specifically, the Mr buffers RBF(j) (j=0, 1, . . . , Mr−1)are updated from the values of RBF(j) to the values of RBF(j+1), and thevalue held in the buffer RBF(0) so far is erased. In STEP4-5, the bufferRBF(Mr) is updated to the target value newly set in STEP4-2 or STEP43.The values of the buffers RBF(0), RBF(1), . . . , RBF(Mr) thus updatedcorrespond respectively to R(n), R(n+1), R(n+Mr) in the fourth term ofthe equation (24). The values of the buffers RBF(0), RBF(1), . . . ,RBF(Mr) have been initialized to a predetermined value (e.g., the targetvalue set in STEP4-2) at the time the engine 1 has started to operate.

The processing in STEP5 is carried out by the heater controller 22 asshown in FIG. 10. The heater controller 22 calculates an elementtemperature difference e(n)=T_(O2)(n)−RBF(0) between the presentestimated value T_(O2)(n) of the element temperature T_(O2) and thevalue of the buffer RBF(0) (=R(n)), i.e., the target value R set by thetarget value setting means 21 prior to the target value predicting timeMr in STEP5-1.

Then, the heater controller 22 determines the values of flags F/A, F/Bin STEP5-2. The flag F/A is set to “0” or “1” in a limiting process(described later on) for limiting the duty cycle DUT. The flag F/A whichis set to “1” means that the duty cycle DUT is forcibly limited to apredetermined upper or lower limit value, and the flag F/B which is setto “0” means that the duty cycle DUT is not limited to the predeterminedupper or lower limit value (the upper limit value>DUT>the lower limitvalue). The flag F/B is set to “1” when the heater 13 is forciblyde-energized by the overheating prevention process. The flags F/A, F/Bare initially set to “0”.

If F/A=F/B=0 in STEP5-2, then the heater controller 22 adds the presentvalue of Σe(j) in the first term of the equation (24) to the differencee(n) calculated in STEP5-1 in STEP5-3. In this manner, the differencee(n) is cumulatively added (integrated) in each cycle time dtc of theprocessing sequence of the heater controller 22. The initial value ofΣe(j) is “0”.

If F/A=1 or F/B=1 in STEP5-2, then since the present value of the dutycycle DUT is not a normal value, the heater controller 22 skips theprocessing in STEP5-3, but goes to STEP5-4, holding the present value ofΣe(j).

Then, the heater controller 22 calculates the equation (24) using thepresent value (latest value) of the element temperature difference e(n)determined in STEP5-2 and the present accumulated value of Σe(j), thuscalculating the present value DUT(n) of the control input DUT for theheater 13 in STEP5-4. Specifically, the heater controller 22 calculatesthe duty cycle DUT(n) according to the equation (24) from the presentvalue of the difference e(n) determined in STEP5-1, the presentaccumulated value Ze(j), the present estimated value Tht(n) of theheater temperature Tht, the present values (=R(n), R(n+1), R(n+Mr)) ofthe buffers RBF(0), RBF(1), . . . , RBF(Mr), the present estimated valueTgd(n) of the exhaust gas temperature Tgd (the exhaust gas temperatureat the location of the O₂ sensor 8), and the values of predeterminedcoefficients Fs0, Fe0, Fx0, Fr(i) (i=0, 1, . . . , Mr), Fdt. When theengine 1 starts to operate, the estimated value of the heatertemperature Tht and the estimated value of the exhaust gas temperatureTgd are set to the atmospheric temperature T_(A) as an initial valuewhich is detected at the start of the engine 1. These initial values ofthe heater temperature Tht and the exhaust gas temperature Tgd are usedin the calculation of the equation (24) when the processing sequences ofexhaust temperature observer 19 and the element temperature observer 20are not executed. After the processing sequences of exhaust temperatureobserver 19 and the element temperature observer 20 are executed, thelatest estimated values determined in the processing sequences ofexhaust temperature observer 19 and the element temperature observer 20are used in the calculation of the equation (24).

Then, the heater controller 22 carries out a limiting process forlimiting the duty cycle DUT(n) calculated in STEP5-4 in STEP5-5 throughSTEP5-11. Specifically, the heater controller 22 determines whether theduty cycle DUT(n) is smaller than a predetermined lower limit value(e.g., “0”) or not in STEP5-5. If DUT(n)<the lower limit value, then theheater controller 22 forcibly sets the value of DUT(n) to the lowerlimit value in STEP5-6. Thereafter, the value of the flag F/A (the flagused in STEP5-2) is set to “1” in STEP5-7.

If DUT(n)≧the lower limit value, then the heater controller 22determines whether the duty cycle DUT(n) is greater than a predeterminedupper limit value (e.g., 100%) or not in STEP5-8. If DUT(n)>the upperlimit value, then the heater controller 22 forcibly sets the value ofDUT(n) to the upper limit value in STEP5-9. Thereafter, the value of theflag F/A is set to “1” in STEP5-10. If the lower limit value≦DUT(n)≦theupper limit value, then the heater controller 22 holds the value ofDUT(n), and sets the flag F/A to “0” in STEP5-11. The processing inSTEP5 is not finished.

Control then returns to the main routine shown in FIG. 8. The sensortemperature control means 18 carries out the processing in STEP7 throughSTEP13. The processing in STEP7 through STEP13 represents a process ofpreventing the heater 13 from being overheated. In STEP7, the sensortemperature control means 18 determines whether or not the presentestimated value (latest value) of the heater temperature Tht is equal toor higher than a predetermined upper limit value THTLMT (e.g., 930° C.).In the present embodiment, if Tht≧THTLMT, the sensor temperature controlmeans 18 forcibly de-energizes the heater 13 to prevent the heater 13from being damaged. However, the estimated value of Tht may temporarilyrise to a value equal to or higher than the upper limit value THTLMT dueto a disturbance or the like. According to the present embodiment,therefore, the sensor temperature control means 18 forcibly de-energizesthe heater 13 if the state in which Tht≧THTLMT has continued for apredetermined time (e.g., 3 seconds, hereinafter referred to as “heaterOFF delay time”).

If Tht<THTLMT in STEP7, then the sensor temperature control means 18sets the value of a countdown timer TMHTOFF for measuring the heater OFFdelay time to a predetermined value TM2 corresponding to the heater OFFdelay time in STEP8. Since the sensor temperature control means 18 doesnot forcibly de-energize the heater 13 at this time, the sensortemperature control means 18 sets the value of the flag F/B (the flagused in STEP5-2 shown in FIG. 10) to “0” in STEP9.

If Tht≧THTLMT in STEP7, then the sensor temperature control means 18counts down the value of the countdown timer TMHTOFF by “1” in STEP10.Then, the sensor temperature control means 18 determines whether thevalue of the countdown timer TMHTOFF is “0” or not, i.e., whether theheater OFF delay time TM2 has elapsed with Tht>THTLMT or not in STEP11.

If TMHTOFF≠0, then the sensor temperature control means 18 sets the flagF/B to “0” in STEP9. If TMHTOFF=0, then the sensor temperature controlmeans 18 forcibly sets the present value of the duty cycle DUT to “0” inSTEP12, and then sets the value of the flag F/B to “1” in STEP13.

When the flag F/B is set to “0” in STEP9, the sensor temperature controlmeans 18 applies a pulsed voltage to the heater energization circuitaccording to the present value of the duty cycle DUT (the latest valuecalculated in STEP5), energizing the heater 13 with the electric energydepending on the duty cycle DUT. When the value of the flag F/B is setto “1” in STEP12, the sensor temperature control means 18 does not applya pulsed voltage to the heater energization circuit, thus de-energizingthe heater 13.

After having thus executed the processing in STEP7 through STEP13, i.e.,the process of preventing the heater 13 from being overheated, thesensor temperature control means 18 determines the value of a countdowntimer COBS for measuring the time dt of one period of the processingsequences of the exhaust temperature observer 19 and the elementtemperature observer 20 in STEP14. The value of the countdown timer COBSis initially set to “0” when the engine 1 has started to operate.

If COBS=0, then the sensor temperature control means 18 newly sets thevalue of COBS to a timer setting time TM3 (shorter than TM1 in STEP3)which corresponds to the period dt of the processing sequences of theexhaust temperature observer 19 and the element temperature observer 20in STEP15. Then, the exhaust temperature observer 19 carries out aprocess of estimating the exhaust gas temperature Tgd (the exhaust gastemperature in the vicinity of the location of the O₂ sensor 8), and theelement temperature observer 20 carries out a process of estimating theelement temperature T_(O2) (including a process of estimating the heatertemperature Tht) in STEP16. If COBS≠0 in STEP14, the exhaust temperatureobserver 19 skips the processing in STEP15 and STEP16. The processing inSTEP16 is therefore carried out at a period dt which is determined bythe timer setting time TM3. The main routine shown in FIG. 8 is nowfinished.

The processing in STEP16 is specifically carried out as shown in FIG.11. The exhaust temperature observer 19 successively carries out theprocessing in STEP16-1 through STEP16-6 to determine an estimated valueof the exhaust gas temperature Tgd in the vicinity of the location ofthe O₂ sensor 8. In STEP16-1, the exhaust temperature observer 19determines a gas speed parameter Vg according to the equation (7) usingthe present detected values (the latest values acquired in STEP1) of therotational speed NE and the intake pressure PB of the engine 1. The gasspeed parameter Vg is forcibly set to Vg=1 if the result calculated bythe equation (7) exceeds “1” due to an excessive rotational speed of theengine 1.

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Texg at the exhaust port 2 of the engine1 according to the equation (1) in STEP16-2. Specifically, the exhausttemperature observer 19 determines a basic exhaust gas temperatureTMAP(NE,PB) from the present detected values of the rotational speed NEand the intake pressure PB of the engine 1 based on the predeterminedmap, and thereafter calculates the right side of the equation (1) usingthe basic exhaust gas temperature TMAP(NE,PB), the present estimatedvalue Texg(k−1) (determined in STEP16-2 in the preceding cycle time) ofthe exhaust gas temperature Texg, and the value of a predeterminedcoefficient Ktex, thus calculating a new estimated value Texg(k) of theexhaust gas temperature Texg. In the present embodiment, while theengine 1 is idling and also while the supply of fuel to the engine 1 isbeing cut off, the basic exhaust gas temperature TMAP used in thecalculation of the equation (1) is set to predetermined valuescorresponding to the respective engine operating states. When the engine1 starts to operate, the atmospheric temperature T_(A) detected at thistime is set as an initial value Texg(0) of the estimated value of theexhaust gas temperature Texg. When the equation (1) is calculated forthe first time after the engine 1 has started to operate, the initialvalue Texg(0) is used as the value of Texg(k−1).

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Tga and an estimated value of the exhaustpipe temperature Twa in the partial exhaust passageway 3 a according tothe respective equations (5-1), (5-2) in STEP16-3. Specifically, theexhaust temperature observer 19 determines a new estimated valueTga(k+1) of the exhaust gas temperature Tga by calculating the rightside of the equation (5-1) using the present estimated value Tga(k)(determined in STEP16-3 in the preceding cycle time) of the exhaust gastemperature Tga, the present estimated value (determined in STEP16-3 inthe preceding cycle time) of the exhaust pipe temperature Twa, thepresent estimated value of the exhaust gas temperature Texg previouslycalculated in STEP16-2, the present value of the gas speed parameter Vgcalculated in STEP16-1, the value of the predetermined model coefficientAa, and the value of the period dt of the processing sequence of theexhaust temperature observer 19.

The exhaust temperature observer 19 calculates a new estimated valueTwa(k+1) of the exhaust pipe temperature Twa by calculating the rightside of the equation (5-2) using the present estimated value Tga(k)(determined in STEP16-3 in the preceding cycle time) of the exhaust gastemperature Tga, the present estimated value (determined in STEP16-3 inthe preceding cycle time) of the exhaust pipe temperature Twa, thevalues of the predetermined model coefficients Ba, Ca, and the value ofthe period dt of the processing sequence of the exhaust temperatureobserver 19.

When the engine 1 starts to operate, the atmospheric temperature T_(A)detected at this time is set as initial values Tga(0), Twa(0) of theestimated values of the exhaust gas temperature Tga and the exhaust pipetemperature Twa. When the equations (5-1), (5-2) are calculated for thefirst time after the engine 1 has started to operate, these initialvalues Tga(0), Twa(0) are used as the respective values of Tga(k−1),Twa(k−1).

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Tgb and an estimated value of the exhaustpipe temperature Twb in the partial exhaust passageway 3 b according tothe respective equations (6-1), (6-2) in STEP16-4. Specifically, theexhaust temperature observer 19 determines a new estimated valueTgb(k+1) of the exhaust gas temperature Tgb by calculating the rightside of the equation (6-1) using the present estimated value Tgb(k)(determined in STEP16-4 in the preceding cycle time) of the exhaust gastemperature Tgb, the present estimated value (determined in STEP16-4 inthe preceding cycle time) of the exhaust pipe temperature Twb, thepresent estimated value of the exhaust gas temperature Tga previouslycalculated in STEP16-3, the present value of the gas speed parameter Vgcalculated in STEP16-1, the value of the predetermined model coefficientAb, and the value of the period dt of the processing sequence of theexhaust temperature observer 19.

The exhaust temperature observer 19 calculates a new estimated valueTwb(k+1) of the exhaust pipe temperature Twb by calculating the rightside of the equation (6-2) using the present estimated value Tgb(k)(determined in STEP16-4 in the preceding cycle time) of the exhaust gastemperature Tgb, the present estimated value (determined in STEP16-4 inthe preceding cycle time) of the exhaust pipe temperature Twb, thevalues of the predetermined model coefficients Bb, Cb, and the value ofthe period dt of the processing sequence of the exhaust temperatureobserver 19.

When the engine 1 starts to operate, the atmospheric temperature T_(A)detected at this time is set as initial values Tgb(0), Twb(0) of theestimated values of the exhaust gas temperature Tgb and the exhaust pipetemperature Twb. When the equations (6-1), (6-2) are calculated for thefirst time after the engine 1 has started to operate, these initialvalues Tgb(0), Twb(0) are used as the respective values of Tgb(k−1),Twb(k−1).

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Tgc and an estimated value of the exhaustpipe temperature Twc in the partial exhaust passageway 3 c according tothe respective equations (8-1), (8-2) in STEP16-5. Specifically, theexhaust temperature observer 19 determines a new estimated valueTgc(k+1) of the exhaust gas temperature Tgc by calculating the rightside of the equation (8-1) using the present estimated value Tgc(k)(determined in STEP16-5 in the preceding cycle time) of the exhaust gastemperature Tgc, the present estimated value (determined in STEP16-5 inthe preceding cycle time) of the exhaust pipe temperature Twc, thepresent estimated value of the exhaust gas temperature Tgb previouslycalculated in STEP16-4, the present value of the gas speed parameter Vgcalculated in STEP16-1, the value of the predetermined model coefficientAc, and the value of the period dt of the processing sequence of theexhaust temperature observer 19.

The exhaust temperature observer 19 calculates a new estimated valueTwc(k+1) of the catalyst temperature Twc by calculating the right sideof the equation (8-2) using the present estimated value Tgc(k)(determined in STEP16-5 in the preceding cycle time) of the exhaust gastemperature Tgc, the present estimated value (determined in STEP16-5 inthe preceding cycle time) of the catalyst temperature Twc, the presentvalue of the gas speed parameter Vg calculated in STEP16-1, the valuesof the predetermined model coefficients Bc, Cc, Dc, and the value of theperiod dt of the processing sequence of the exhaust temperature observer19.

When the engine 1 starts to operate, the atmospheric temperature T_(A)detected at this time is set as initial values Tgc(0), Twc(0) of theestimated values of the exhaust gas temperature Tgc and the exhaust pipetemperature Twc. When the equations (8-1), (8-2) are calculated for thefirst time after the engine 1 has started to operate, these initialvalues Tgc(0), Twc(0) are used as the respective values of Tgc(k−1),Twc(k−1).

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Tgd and an estimated value of the exhaustpipe temperature Twd in the partial exhaust passageway 3 d (near thelocation of the O₂ sensor 8) according to the respective equations(9-1), (9-2) in STEP16-6. Specifically, the exhaust temperature observer19 determines a new estimated value Tgd(k+1) of the exhaust gastemperature Tgd by calculating the right side of the equation (9-1)using the present estimated value Tgd(k) (determined in STEP16-6 in thepreceding cycle time) of the exhaust gas temperature Tgd, the presentestimated value (determined in STEP16-6 in the preceding cycle time) ofthe exhaust pipe temperature Twd, the present estimated value of theexhaust gas temperature Tgc previously calculated in STEP16-5, thepresent value of the gas speed parameter Vg calculated in STEP16-1, thevalue of the predetermined model coefficient Ad, and the value of theperiod dt of the processing sequence of the exhaust temperature observer19.

The exhaust temperature observer 19 calculates a new estimated valueTwd(k+1) of the exhaust pipe temperature Twd by calculating the rightside of the equation (9-2) using the present estimated value Tgd(k)(determined in STEP16-6 in the preceding cycle time) of the exhaust gastemperature Tgd, the present estimated value (determined in STEP16-6 inthe preceding cycle time) of the exhaust pipe temperature Twd, thevalues of the predetermined model coefficients Bd, Cd, and the value ofthe period dt of the processing sequence of the exhaust temperatureobserver 19.

When the engine 1 starts to operate, the atmospheric temperature T_(A)detected at this time is set as initial values Tgd(0), Twd(0) of theestimated values of the exhaust gas temperature Tgd and the exhaust pipetemperature Twd. When the equations (9-1), (9-2) are calculated for thefirst time after the engine 1 has started to operate, these initialvalues Tgd(0), Twd(0) are used as the respective values of Tgd(k−1),Twd(k−1).

Then, the element temperature observer 20 executes the processing inSTEP16-7 to determine estimated values of the element temperature T_(O2)of the O₂ sensor 8 and the heater temperature Tht according to theequations (10-1), (10-2). Specifically, the element temperature observer20 determines a new estimated value T_(O2)(k+1) of the devicetemperature T_(O2) by calculating the right side of the equation (10-1)using the present estimated value T_(O2)(k) (determined in STEP16-7 inthe preceding cycle time) of the element temperature T_(O2), the presentestimated value Tht(k) (determined in STEP16-7 in the preceding cycletime) of the heater temperature Tht, the present estimated value of theexhaust gas temperature Tgd previously calculated in STEP16-6, thevalues of the predetermined model coefficients Ax, Bx, and the value ofthe period dt (=the period of the of the processing sequence of theexhaust temperature observer 19) of the processing sequence of theelement temperature observer 20.

Then, the element temperature observer 20 determines a new estimatedvalue Tht(k+1) of the heater temperature Tht by calculating the rightside of the equation (10-2) using the present estimated value T_(O2)(k)(determined in STEP16-7 in the preceding cycle time) of the elementtemperature T_(O2), the present estimated value Tht(k) (determined inSTEP16-7 in the preceding cycle time) of the heater temperature Tht, thepresent value DUT(k) of the duty cycle DUT, the values of thepredetermined model coefficients Cx, Dx, and the value of the period dtof the processing sequence of the element temperature observer 20.

When the engine 1 starts to operate, the atmospheric temperature T_(A)detected at this time is set as initial values T_(O2)(0), Tht(0) of theestimated values of the element temperature T_(O2) and the heatertemperature Tht. When the equations (10-1), (10-2) are calculated forthe first time after the engine 1 has started to operate, these initialvalues T_(O2)(0), Tht(0) are used as the respective values ofT_(O2)(k−1), Tht(k−1). The duty cycle DUT(k) used in the equation (10-2)is basically of the latest value determined by the heater controller 22in STEP5. However, if the value of the duty cycle DUT is limited inSTEP12 to de-energize the heater 13, then the limited value of the dutycycle DUT is used in the equation (10-2).

The above processing sequence of the sensor temperature control means 18controls the electric energy supplied to the heater 13 of the O₂ sensor8 in order to keep the element temperature T_(O2) of the O₂ sensor 8 atthe target value R. Except immediately after the engine 1 has started tooperate and when the atmospheric temperature T_(A) is considerably low,the target value R is normally set to 800° C. As a result, the outputcharacteristics of the O₂ sensor 8 can be maintained stably as thecharacteristics suitable for controlling the air-fuel ratio of theengine 1, i.e., for controlling the air-fuel ratio thereof for thecatalytic converter 4 to perform a better exhaust purifying capability,and the air-fuel ratio of the engine 1 can well be controlled to allowthe catalytic converter 4 to perform a better exhaust purifyingcapability.

According to the present embodiment, the temperature T_(O2) of the O₂sensor 8 and the heater temperature Tht are estimated by the elementtemperature observer 20. At this time, the temperature T_(O2) issequentially estimated based on the model equation (10-1) which isconstructed in view of a temperature change depending on the heattransfer between the active element 10 and the heater 13 and atemperature change depending on the heat transfer between the activeelement 10 and the exhaust gas in the vicinity of the location where theO₂ sensor 8 is located (the exhaust gas held in contact with the activeelement 10). The heater temperature Tht is sequentially estimated basedon the model equation (10-2) which is constructed in view of atemperature change depending on the heat transfer between the activeelement 10 and the heater 13 and a temperature change due to theelectric power supplied to the heater 13, i.e., a temperature changedepending on the duty cycle DUT which determines the amount of electricpower supplied to the heater 13.

As a result, the element temperature T_(O2) and the heater temperatureTht can be estimated in a manner taking into account the application ofheat to the active element 10 and the heater 13 appropriately, so thatthe estimated values of those temperatures have a sufficiently highlevel of accuracy.

The duty cycle DUT as a control input for the heater 13 is calculated asincluding a control input component (the first term (the term includingΣe(j)) and the second term (the term including e(n)) of the equation(24)) depending on the estimated value of the element temperature T_(O2)and also a control input component (the third term of the equation (24)depending on the estimated value of the heater temperature Tht.Therefore, combined with the fact that the accuracy of the estimatedvalues of the element temperature T_(O2) and the heater temperature Thtis sufficiently high, the element temperature T_(O2) can stably becontrolled reliably at a desired target value R.

According to the present embodiment, the duty cycle DUT is calculated asincluding, in addition to a control input component depending on theelement temperature T_(O2) and the heater temperature Tht, a controlinput component depending on the estimated value of the exhaust gastemperature Tgd which acts as a disturbant factor for varying theelement temperature T_(O2), i.e., the optimum disturbance value F/Fcomponent Uopfd. The coefficient Fdt relative to the optimum disturbancevalue F/F component Uopfd is determined according to a predictivecontrol algorithm on the assumption that the present exhaust gastemperature will continue until after the exhaust gas temperaturepredicting time Md. As a result, the stability of the process ofcontrolling the element temperature T_(O2) at the target value R iseffectively increased, and hence the stability of the outputcharacteristics of the O₂ sensor 8 is also effectively increased.

According to the present embodiment, furthermore, the control input DUTis calculated as including the control input component depending on thetarget value R for the element temperature T_(O2) (the target value Rfrom the present until after the target value predicting time Mr), i.e.,the optimum target value F/F component Uopfr. When the target value Rchanges from a low temperature (600° C.) immediately after the engine 1has started to operate to a normal high temperature (750° C. through800° C.) in particular, the control input DUT is prevented from becomingtemporarily large excessively, i.e., the element temperature T_(O2) isprevented from overshooting with respect to the target value R. Thestability of the output characteristics of the O₂ sensor 8 is alsoeffectively increased.

A second embodiment of the present invention will be described belowwith reference to FIG. 12. The second embodiment is partly different inarrangement or function from the first embodiment described above, andthose structural or functional parts of the second embodiment which areidentical to those of the first embodiment are denoted by identicalreference characters, and will not be described in detail below.

According to the present embodiment, as shown in the block diagram ofFIG. 12, the sensor temperature control means 18 of the control unit 16shown in FIG. 1 comprises, as functional means, an exhaust temperatureobserver 19, an element temperature observer 20, a target value settingmeans 31, and a heater controller 32. The exhaust temperature observer19 and the element temperature observer 20 are identical to those of thefirst embodiment. According to the present embodiment, however, theelement temperature observer 20 corresponds to a heater temperatureestimating means according to the present invention. In the presentembodiment, the target value setting means 31 and the heater controller32 have their processing periods identical to the processing periods ofthe target value setting means 21 and the heater controller 22 accordingthe first embodiment.

The target value setting means 31 serves to set a target value R′ forthe heater temperature Tht of the O₂ sensor 8. According to theinventors' knowledge, the heater temperature Tht is relatively highlycorrelated to the element temperature T_(O2) and tends to be higher thanthe element temperature T_(O2) by a constant temperature in a steadystate. According to the present embodiment, the target value settingmeans 31 sets, as the target value R′ for the heater temperature Tht, avalue R+DR which is higher than the target value R for the elementtemperature T_(O2) that is set as described in the first embodiment (thetarget value R set by the processing sequence shown in FIG. 9), by apredetermined value DR (e.g., 100° C.). As with the first embodiment,the target value R′ that is set by the target value setting means 31 ineach cycle time of its processing sequence is a target value after thetarget value predicting time Mr, and the target value R′ in the periodof the target value predicting time Mr is sequentially updated andstored.

The heater controller 32 sequentially generates the duty cycle DUT as acontrol input in order to keep the heater temperature Tht at the targetvalue R′. In the present embodiment, as with the first embodiment, theheater controller 32 calculates the duty cycle DUT according to anoptimum predictive control algorithm.

More specifically, according to the present embodiment, attention ispaid to the difference e′ between the heater temperature Tht and atarget value R′ therefor, a change Δe′ per given time in the differencee′ (corresponding to a rate of change of the difference e′), and achange AT_(O2) per given time in the element temperature T_(O2)(corresponding to a rate of change of the element temperature T_(O2)),and a model equation for an object to be controlled by the heatercontroller 32 is introduced using the above differences and changes asstate quantities relative to the object to be controlled by the heatercontroller 32.

If the difference e′ (hereinafter referred to as “heater temperaturedifference e′”) is defined as e′(n)=Tht(n)−R′(n), then the modelequation is given as the following equation (25) based on the aboveequations (11-1), (11-2) according to the same idea as with the firstembodiment:X1(n+1)=Φ′·X1(n)+G′·ΔDUT(n)+Gd′·ΔTgd(n)+Gr′R1(n+1)  (25)where

-   X1(n)=(e′(n),Δe′(n),ΔT_(O2)(n))^(T),-   R1(n+1)=(ΔR′(n+1),ΔR′(n))^(T),-   G′=(0,Dx·dtc,0)^(T),-   Gd′=(0,0,Ax·dtc)^(T),

$\Phi^{\prime} = \begin{bmatrix}1 & 1 & 0 \\0 & {1 - {{Cx} \cdot {dtc}}} & {{Cx} \cdot {dtc}} \\0 & {{Bx} \cdot {dtc}} & {1 - {{Ax} \cdot {dtc}} - {{Bx} \cdot {dtc}}}\end{bmatrix}$ ${Gr}^{\prime} = \begin{bmatrix}0 & 0 \\{- 1} & {1 - {{Cx} \cdot {dtc}}} \\0 & {{Bx} \cdot {dtc}}\end{bmatrix}$

In the present embodiment, the control input DT to be determined by theheater controller 32 is given by the equation (27) shown below as havingintegrated ADUT which minimizes an evaluating function J1 according tothe following equation (26):

$\begin{matrix}{{{J1} = {\sum\limits_{n = {M + 1}}^{\infty}\left\lbrack {{{{X1}^{T}(n)} \cdot {Q0} \cdot {{X1}(n)}} + {\Delta\;{{{DUT}^{T}(n)} \cdot {H0} \cdot \Delta}\;{{DUT}(n)}}} \right\rbrack}}{{{where}\mspace{14mu} M} = {\max\mspace{14mu}\left( {M,{Md}} \right)}}} & (26) \\{{{DUT}(n)} = {{Fs1} + {\sum\limits_{j = 1}^{n}{e^{\prime}(j)}} + {{Fe1} \cdot {e^{\prime}(n)}} + {{Fx1} \cdot {T_{02}(n)}} + {\sum\limits_{i = 1}^{Mr}\left\lbrack {{{Fr}^{\prime}(i)} \cdot {R^{\prime}\left( {n + 1} \right)}} \right\rbrack} + {{Fdt}^{\prime}{{Tgd}(n)}}}} & (27)\end{matrix}$

The coefficients Fs1, Fe1, Fx1 in the first through third terms, thecoefficient Fr′(i) (i=0, 1, . . . , Mr) in the fourth term, and thecoefficient Fdt′ in the fifth term on the right side of the equation(27) are coefficients given respectively by the following equations(28-1) through (28-3):

$\begin{matrix}\begin{matrix}{{F1} \equiv \left( {{Fs1},{Fe1},{Fx1}} \right)} \\{= {{- \left\lbrack {{H0} + {G^{\prime\; T} \cdot P^{\prime} \cdot G^{\prime}}} \right\rbrack^{- 1}} \cdot G^{\prime\; T} \cdot P^{\prime} \cdot \Phi^{\prime}}}\end{matrix} & \left( {28\text{-}1} \right) \\{{{Fr}^{\prime}(i)} = \left\lbrack \begin{matrix}{{{Fr12}(1)}\text{:}} & {i = 0} \\{{{Fr11}(i)} + {{{Fr12}\left( {i + 1} \right)}\text{:}}} & {{i = 1},2,\ldots\mspace{11mu},{{Mr} - 1}} \\{{{Fr11}({Mr})}\text{:}} & {i = {Mr}}\end{matrix} \right.} & \left( {28\text{-}2} \right) \\{{Fdt}^{\prime} = {\sum\limits_{i = 0}^{Md}\left\{ {{- \left\lbrack {{H0} + {G^{\prime\; T} \cdot P^{\prime} \cdot G^{\prime}}} \right\rbrack^{- 1}} \cdot G^{\prime\; T} \cdot \left( \zeta^{\; T} \right)^{i} \cdot P^{\prime} \cdot {Gd}^{\prime}} \right\}}} & \left( {28\text{-}3} \right)\end{matrix}$where

-   P′=Q0+Φ′^(T) ·P′·Φ′, −Φ′·P′·G′·[H 0+G′^(T) ·P′·G′] ⁻¹ ·G′ ^(T)    ·P′·Φ′-   ζ′=Φ′+G′·F1-   (Fr11(i), Fr12(i))=[H0+G′ ^(T) ·P′·G′] ⁻¹·G′·^(T)·(ζ′^(t))^(i−1)    ·P′·Gr′(i=1, 2, . . . , Mr)

In the present embodiment, the weighted matrixes Q0, H0 with respect tothe evaluating function J1, the target value predicting time Mr, and theexhaust gas temperature predicting time Md are identical to those in thefirst embodiment. However, they may be set to values different fromthose in the first embodiment. The coefficients Fs1, Fe1, Fx1, Fr′(i),Fdt′ in the equation (27) may not necessarily be of the values accordingto the defining equations (28-1) through (28-3), but may be of valuesadjusted by way of simulation or experimentation. Furthermore, thecoefficients Fs1, Fe1, Fx1, Fr′(i), Fdt′ may be changed depending on theelement temperature, the heater temperature, etc. In the presentembodiment, as with the first embodiment, the exhaust gas temperatureTgd is maintained at the present value in the future until after Mdsteps. However, if Tgd at each time in the future can be detected orestimated, then the control input DUT may be determined using thosevalues (in this case, Fdt′ is a vector).

The above equation (27) is a formula for sequentially calculating acontrol input DUT(n) (duty cycle) with which the heater controller 32controls the heater 13 in the present embodiment. Specifically, theheater controller 32 sequentially calculates the control input DUT(n) ineach cycle time (period) of the control processing of the heatercontroller 32 according to the equation (27), and applies a pulsevoltage having the duty cycle DUT(n) to a heater energizing circuit, notshown, thereby adjusting the electric power supplied to the heater 13.The terms on the right side of the equation (27) have the same meaningsas those in the first embodiment. Specifically, the first through thirdterms (the term including Σe′(j) through the term including T_(O2)(n))on the right side represent a control input component (a feedbackcomponent based on an optimum control algorithm) depending on the heatertemperature difference e′ and the element temperature T_(O2). The fourthterm (the term of ΣFr′(i)·R′(n+i)) on the right side of the equation(24) and the fifth term (the term including Tgd(n)) on the right sidethereof represent a control input component (a feed-forward componentbased on a predictive control algorithm) depending on the exhaust gastemperature Tgd.

As the element temperature T_(O2) and the exhaust gas temperature Tgdthat are required to determine the control input DUT(n) according to theequation (27), there are employed, respectively, the latest value of theestimated value of the element temperature T_(O2) determined by theelement temperature observer 20 and the latest value of the estimatedvalue of the exhaust gas temperature Tgd determined by the exhausttemperature observer 19.

The heater temperature difference e′ required for the calculationaccording to the equation (27) is calculated from the latest value ofthe estimated value of the heater temperature Tht determined by theelement temperature observer 20 and the target value R′ that has beenset in a cycle time before the target value predicting time Mr by thetarget value setting means 31.

The other processing details than those described above are identical tothose according to the first embodiment. In the present embodiment, theelectric power supplied to the heater 13 of the O₂ sensor 8 iscontrolled in order to maintain the heater temperature Tht of the O₂sensor 8 at the target value R′. In this case, except immediately afterthe engine 1 starts to operate or when the atmospheric temperature T_(A)is considerably low, the target value R′ is usually set to a temperature(900° C.) which is higher than a preferred target temperature of 800° C.for the active element 10 by a predetermined value DR (100° C. in thepresent embodiment). As a result, the temperature T_(O2) of the activeelement 10 of the O₂ sensor 8 is indirectly controlled substantially atthe temperature of 800° C. Therefore, as with the first embodiment, theoutput characteristics of the O₂ sensor 8 can stably be kept ascharacteristics suitable for controlling the air-fuel ratio of theengine 1 (for controlling the air-fuel ratio to keep a good purifyingcapability of the catalytic converter 4), and hence the air-fuel ratiois controlled well to reliably keep a good purifying capability of thecatalytic converter 4. During a predetermined period of time immediatelyafter the engine 1 has started to operate, the target temperature R′ forthe heater 13 is set to a temperature (700° C.) which is higher than alow temperature (600° C.) as the target temperature R for the activeelement 10 than the predetermined value DR, for thereby preventing theactive element 10 from being damaged by stresses due to abrupt heating.If the atmospheric temperature T_(A) is low (T_(A)<0° C.), then inasmuchas the target value R for the active element 10 is set to a value in therange of 750° C.≦R<800° C., the target value R′ for the heater 13 is setto a value in the range of 850° C.≦R′<900° C. to prevent the heater 13from being overheated.

According to the present embodiment, since the algorithm of the elementtemperature observer 20 is the same as the algorithm according to thefirst embodiment, the accuracy of the estimated values of the elementtemperature T_(O2) and the heater temperature Tht is sufficientlymaintained. The duty cycle DUT as a control input to the heater 13includes the control input component (the first term (the term includingΣe′(j)) and the second term (the term including e′(n)) of the equation(27)) depending on the estimated value of the heater temperature Tht,and the control input component (the third term of the equation (27))depending on the estimated value of the element temperature T_(O2). Inaddition, according to the present embodiment, a predictive controlalgorithm is also applied, and the duty cycle DUT includes the controlinput component (a feed-forward component of the fifth term on the rightside of the equation (27)) depending on the exhaust gas temperature Tgd,and the control input component (a feed-forward component of the fourthterm on the right side of the equation (27)) depending on the targetvalue R′. As a result, the heater temperature Tht can reliably becontrolled stably at the desired target value R′, and the elementtemperature T_(O2) can be controlled stably at a desired temperature.

In the first and second embodiments described above, the exhaust gastemperature Tgd is estimated. However, an exhaust gas sensor may bedisposed in the vicinity of the O₂ sensor 8, and the exhaust gastemperature Tgd may be detected by the exhaust gas sensor. In this case,the element temperature Tht is estimated using the detected value(latest value) of the exhaust gas temperature sensor as the value of theexhaust gas temperature Tgd in the equation (10-1). The duty cycle DUTmay be calculated using the detected value (latest value) of the exhaustgas temperature sensor as the value of the exhaust gas temperature Tgdin the equation (24) or (27).

In the first and second embodiments, both the element temperature T_(O2)and the heater temperature Tht are estimated. However, either one ofthem may be detected directly by a temperature sensor. If the elementtemperature T_(O2) is detected, then the heater temperature Tht may beestimated using the detected value (latest value) of the elementtemperature T_(O2) as the value of the element temperature T_(O2) in theequation (10-2), and the duty cycle DUT may be calculated using thedetected value (latest value) of the element temperature T_(O2) as thevalue of the element temperature T_(O2) in the equation (27). If theheater temperature Tht is detected, then the element temperature T_(O2)may be estimated using the detected value (latest value) of the heatertemperature Tht as the value of the heater temperature Tht in theequation (10-1), and the duty cycle DUT may be calculated using thedetected value (latest value) of the heater temperature Tht as the valueof the heater temperature Tht in the equation (24).

In the first and second embodiments, the element temperature T_(O2) ofthe O₂ sensor 8 or the heater temperature Tht is controlled at thetarget value R or R′ according to the optimum predictive controlalgorithm. However, the control input DUT may be generated according toanother control algorithm (e.g., an ordinary PI or PID control process).

Alternatively, the control input DUT may be determined according to anordinary optimum control algorithm which includes no predictive controlalgorithm. In this case, the control input DUT may sequentially becalculated according to an equation which is produced by removing thefourth term (the term including R(n+i)) and the fifth term (the termincluding Tgd(n)) from the equation (24) or by removing the fourth term(the term including R′(n+i)) and the fifth term (the term includingTgd(n)) from the equation (27). According to this modification, theheater controller for determining the control input DUT is an optimumservocontroller for determining the control input DUT in order tominimize the value of the evaluating function JO or J1 where M=0 in theequation (17) or (26).

In each of the embodiments described above, the element temperatureT_(O2) of the O₂ sensor 8 is controlled. However, the present inventionis also applicable to an exhaust gas sensor other than the O₂ sensor 8(e.g., the wide-range air-fuel ratio sensor 9 or a humidity sensor forgenerating an output signal representative of the water content of theexhaust gas). In this case, an algorithm for estimating the elementtemperature or the heater controller may be the same as the algorithm inthe first embodiment and the second embodiment.

The internal combustion engine to which the present invention isapplicable may be an ordinary port-injected internal combustion engine,a spark-ignition internal combustion engine with direct fuel injectioninto cylinders, a diesel engine, an internal combustion engine for useas an outboard engine on a boat, etc.

INDUSTRIAL APPLICABILITY

As described above, the present invention is useful as a technology forappropriately controlling the temperature of an exhaust gas sensordisposed in the exhaust system of an internal combustion engine mountedon an automobile, a hybrid vehicle, an outboard engine assembly, or thelike, at a desirable temperature for stabilizing the outputcharacteristics of the exhaust gas sensor.

1. An apparatus for controlling the temperature of an exhaust gas sensordisposed in an exhaust passage of an internal combustion engine andhaving an active element for contacting an exhaust gas flowing throughthe exhaust passage and a heater for heating the active element,comprising: temperature estimating means for sequentially estimating thetemperature of the active element based on a predetermined elementtemperature model which is representative of a temperature change of theactive element due to heat transfer between at least said active elementand an exhaust gas held in contact with the active element, and heatercontrol means for controlling said heater to equalize the temperature ofthe active element with a predetermined target temperature, using anestimated value of the temperature of the active element from saidtemperature estimating means.
 2. The apparatus for controlling thetemperature of an exhaust gas sensor according to claim 1, wherein saidelement temperature model comprises a model which is determined torepresent, in combination, a temperature change of the active elementdue to heat transfer between said active element and an exhaust gas heldin contact with the active element and a temperature change of theactive element due to heat transfer between said active element and saidheater.
 3. The apparatus for controlling the temperature of an exhaustgas sensor according to claim 2, wherein said element temperature modelcomprises a model which is representative of a change per predeterminedtime in the temperature of said active element as including atemperature change component depending on the difference between thetemperature of the active element and the temperature of the exhaust gasheld in contact with the active element, and a temperature changecomponent depending on the difference between the temperature of theactive element and the temperature of said heater, and said temperatureestimating means sequentially estimates a temperature change of saidactive element based on said element temperature model, andaccumulatively adds an estimated value of the temperature change to aninitial value which is set when said internal combustion engine startsto operate, thereby estimating the temperature of the active element. 4.The apparatus for controlling the temperature of an exhaust gas sensoraccording to claim 3, wherein said initial value is set depending on theatmospheric temperature and/or the temperature of the internalcombustion engine at least when said internal combustion engine startsto operate.
 5. The apparatus for controlling the temperature of anexhaust gas sensor according to claim 1, wherein said elementtemperature model comprises a model which is representative of a changeper predetermined time in the temperature of said active element asincluding a temperature change component depending on the differencebetween at least the temperature of the active element and thetemperature of the exhaust gas held in contact with the active element,and said temperature estimating means sequentially estimates atemperature change of said active element based on said elementtemperature model, and accumulatively adds an estimated value of thetemperature change to an initial value which is set when said internalcombustion engine starts to operate, thereby estimating the temperatureof the active element.
 6. The apparatus for controlling the temperatureof an exhaust gas sensor according to claim 5, wherein said initialvalue is set depending on the atmospheric temperature and/or thetemperature of the internal combustion engine at least when saidinternal combustion engine starts to operate.
 7. The apparatus forcontrolling the temperature of an exhaust gas sensor according to claim1, wherein said heater control means sequentially generates a controlinput which determines an amount of heating energy supplied to saidheater, depending on at least the estimated value of the temperature ofthe active element from said temperature estimating means, and controlssaid heater depending on the control input.
 8. An apparatus forcontrolling the temperature of an exhaust gas sensor disposed in anexhaust passage of an internal combustion engine and having an activeelement for contacting an exhaust gas flowing through the exhaustpassage and a heater for heating the active element, comprising:temperature estimating means for sequentially estimating the temperatureof the active element based on a predetermined element temperature modelwhich is representative of a temperature change of the active elementdue to heat transfer between at least said active element and saidheater, and heater control means for controlling said heater to equalizethe temperature of the active element with a predetermined targettemperature, using an estimated value of the temperature of the activeelement from said temperature estimating means.
 9. The apparatus forcontrolling the temperature of an exhaust gas sensor according to claim8, wherein said element temperature model comprises a model which isrepresentative of a change per predetermined time in the temperature ofsaid active element as including a temperature change componentdepending on the difference between at least the temperature of theactive element and the temperature of said heater, and said temperatureestimating means sequentially estimates a temperature change of saidactive element based on said element temperature model, andaccumulatively adds an estimated value of the temperature change to aninitial value which is set when said internal combustion engine startsto operate, thereby estimating the temperature of the active element.10. The apparatus for controlling the temperature of an exhaust gassensor according to claim 9, wherein said initial value is set dependingon the atmospheric temperature and/or the temperature of the internalcombustion engine at least when said internal combustion engine startsto operate.
 11. The apparatus for controlling the temperature of anexhaust gas sensor according to claim 8, wherein said elementtemperature model comprises a model which is determined to represent, incombination, a temperature change of the active element due to heattransfer between said active element and an exhaust gas held in contactwith the active element and a temperature change of the active elementdue to heat transfer between said active element and said heater. 12.The apparatus for controlling the temperature of an exhaust gas sensoraccording to claim 11, wherein said element temperature model comprisesa model which is representative of a change per predetermined time inthe temperature of said active element as including a temperature changecomponent depending on the difference between the temperature of theactive element and the temperature of the exhaust gas held in contactwith the active element, and a temperature change component depending onthe difference between the temperature of the active element and thetemperature of said heater, and said temperature estimating meanssequentially estimates a temperature change of said active element basedon said element temperature model, and accumulatively adds an estimatedvalue of the temperature change to an initial value which is set whensaid internal combustion engine starts to operate, thereby estimatingthe temperature of the active element.
 13. The apparatus for controllingthe temperature of an exhaust gas sensor according to claim 12, whereinsaid initial value is set depending on the atmospheric temperatureand/or the temperature of the internal combustion engine at least whensaid internal combustion engine starts to operate.
 14. The apparatus forcontrolling the temperature of an exhaust gas sensor according to claim8, wherein said heater control means sequentially generates a controlinput which determines an amount of heating energy supplied to saidheater, depending on at least the estimated value of the temperature ofthe active element from said temperature estimating means, and controlssaid heater depending on the control input.
 15. An apparatus forcontrolling the temperature of an exhaust gas sensor disposed in anexhaust passage of an internal combustion engine and having an activeelement for contacting an exhaust gas flowing through the exhaustpassage and a heater for heating the active element, comprising:temperature estimating means. for sequentially estimating thetemperature of the heater based on a predetermined heater temperaturemodel which is representative of a temperature change of the heater dueto heat transfer between at least said heater and said active element,and heater control means for controlling said heater to equalize thetemperature of the heater with a predetermined target temperature, usingan estimated value of the temperature of the heater from saidtemperature estimating means.
 16. The apparatus for controlling thetemperature of an exhaust gas sensor according to claim 15, wherein saidheater temperature model comprises a model which is determined torepresent, in combination, a temperature change of the heater due toheat transfer between said heater and said active element and atemperature change of the heater due to the supply of heating energy tosaid heater.
 17. The apparatus for controlling the temperature of anexhaust gas sensor according to claim 16, wherein said heatertemperature model comprises a model which is representative of a changeper predetermined time in the temperature of said heater as including atemperature change component depending on the difference between thetemperature of the heater and the temperature of the active element, anda temperature change component depending on an amount of heating energysupplied to said heater, and said temperature estimating meanssequentially estimates a temperature change of said heater based on saidheater temperature model, and accumulatively adds an estimated value ofthe temperature change to an initial value which is set when saidinternal combustion engine starts to operate, thereby estimating thetemperature of the heater.
 18. The apparatus for controlling thetemperature of an exhaust gas sensor according to claim 17, wherein saidinitial value is set depending on the atmospheric temperature and/or thetemperature of the internal combustion engine at least when saidinternal combustion engine starts to operate.
 19. The apparatus forcontrolling the temperature of an exhaust gas sensor according to claim15, wherein said heater temperature model comprises a model which isrepresentative of a change per predetermined time in the temperature ofsaid heater as including a temperature change component depending on thedifference between the temperature of the heater and the temperature ofthe active element, and said temperature estimating means sequentiallyestimates a temperature change of said heater based on said heatertemperature model, and accumulatively adds an estimated value of thetemperature change to an initial value which is set when said internalcombustion engine starts to operate, thereby estimating the temperatureof the heater.
 20. The apparatus for controlling the temperature of anexhaust gas sensor according to claim 19, wherein said initial value isset depending on the atmospheric temperature and/or the temperature ofthe internal combustion engine at least when said internal combustionengine starts to operate.
 21. The apparatus for controlling thetemperature of an exhaust gas sensor according to claim 15, wherein saidheater control means sequentially generates a control input whichdetermines an amount of heating energy supplied to said heater,depending on at least the estimated value of the temperature of theheater from said temperature estimating means, and controls said heaterdepending on the control input.
 22. An apparatus for controlling thetemperature of an exhaust gas sensor disposed in an exhaust passage ofan internal combustion engine and having an active element forcontacting an exhaust gas flowing through the exhaust passage and aheater for heating the active element, comprising: temperatureestimating means for sequentially estimating the temperature of theheater based on a predetermined heater temperature model which isrepresentative of a temperature change of the heater due to at least thesupply of heating energy to said heater, and heater control means forcontrolling said heater to equalize the temperature of the heater with apredetermined target temperature, using an estimated value of thetemperature of the heater from said temperature estimating means. 23.The apparatus for controlling the temperature of an exhaust gas sensoraccording to claim 22, wherein said heater temperature model comprises amodel which is representative of a change per predetermined time in thetemperature of said heater as including a temperature change componentdepending on an amount of heating energy supplied to said heater, andsaid temperature estimating means sequentially estimates a temperaturechange of said heater based on said heater temperature model, andaccumulatively adds an estimated value of the temperature change to aninitial value which is set when said internal combustion engine startsto operate, thereby estimating the temperature of the heater.
 24. Theapparatus for controlling the temperature of an exhaust gas sensoraccording to claim 23, wherein said initial value is set depending onthe atmospheric temperature and/or the temperature of the internalcombustion engine at least when said internal combustion engine startsto operate.
 25. The apparatus for controlling the temperature of anexhaust gas sensor according to claim 22, wherein said heatertemperature model comprises a model which is determined to represent, incombination, a temperature change of the heater due to heat transferbetween said heater and said active element and a temperature change ofthe heater due to the supply of heating energy to said heater.
 26. Theapparatus for controlling the temperature of an exhaust gas sensoraccording to claim 25, wherein said heater temperature model comprises amodel which is representative of a change per predetermined time in thetemperature of said heater as including a temperature change componentdepending on the difference between the temperature of the heater andthe temperature of the active element, and a temperature changecomponent depending on an amount of heating energy supplied to saidheater, and said temperature estimating means sequentially estimates atemperature change of said heater based on said heater temperaturemodel, and accumulatively adds an estimated value of the temperaturechange to an initial value which is set when said internal combustionengine starts to operate, thereby estimating the temperature of theheater.
 27. The apparatus for controlling the temperature of an exhaustgas sensor according to claim 26, wherein said initial value is setdepending on the atmospheric temperature and/or the temperature of theinternal combustion engine at least when said internal combustion enginestarts to operate.
 28. The apparatus for controlling the temperature ofan exhaust gas sensor according to claim 22, wherein said heater controlmeans sequentially generates a control input which determines an amountof heating energy supplied to said heater, depending on at least theestimated value of the temperature of the heater from said temperatureestimating means, and controls said heater depending on the controlinput.
 29. An apparatus for controlling the temperature of an exhaustgas sensor disposed in an exhaust passage of an internal combustionengine and having an active element for contacting an exhaust gasflowing through the exhaust passage and a heater for heating the activeelement, comprising: temperature estimating means for sequentiallyestimating the temperature of the active element based on apredetermined element temperature model which is representative of, incombination, a temperature change of the active element due to heattransfer between said active element and an exhaust gas held in contactwith the active element, and a temperature change of the active elementdue to heat transfer between the active element and said heater, andsequentially estimating the temperature of the heater based on apredetermined heater temperature model which is representative of, incombination, a temperature change of the heater due to heat transferbetween said heater and said active element and a temperature change ofthe heater due to the supply of heating energy to said heater; andheater control means for controlling said heater to equalize thetemperature of the active element with a predetermined targettemperature, using an estimated value of the temperature of the activeelement and an estimated value of the temperature of the heater fromsaid temperature estimating means.
 30. The apparatus for controlling thetemperature of an exhaust gas sensor according to claim 29, wherein saidelement temperature model comprises a model which is representative of achange per predetermined time in the temperature of said active elementas including a temperature change component depending on the differencebetween the temperature of the active element and the temperature of theexhaust gas held in contact with the active element, a temperaturechange component depending on the difference between the temperature ofthe active element and the temperature of said heater; said heatertemperature model comprises a model which is representative of a changeper predetermined time in the temperature of said heater as including atemperature change component depending on the difference between thetemperature of the heater and the temperature of the active element, anda temperature change component depending on an amount of heating energysupplied to said heater; and said temperature estimating meanssequentially estimates a temperature change of said active element basedon said element temperature model, and accumulatively adds an estimatedvalue of the temperature change to an initial value of the temperatureof the active element which is set when said internal combustion enginestarts to operate, thereby estimating the temperature of the activeelement, and sequentially estimates a temperature change of said heaterbased on said heater temperature model, and accumulatively adds anestimated value of the temperature change to an initial value of thetemperature of the heater which is set when said internal combustionengine starts to operate, thereby estimating the temperature of theheater.
 31. The apparatus for controlling the temperature of an exhaustgas sensor according to claim 30, wherein said initial value is setdepending on the atmospheric temperature and/or the temperature of theinternal combustion engine at least when said internal combustion enginestarts to operate.
 32. The apparatus for controlling the temperature ofan exhaust gas sensor according to claim 29, wherein said heater controlmeans sequentially generates a control input which determines an amountof heating energy supplied to said heater by adding an input componentdepending on at least the estimated value of the temperature of theactive element from said temperature estimating means and an inputcomponent depending on at least the estimated value of the temperatureof the heater from said temperature estimating means, and controls saidheater depending on the control input.
 33. An apparatus for controllingthe temperature of an exhaust gas sensor disposed in an exhaust passageof an internal combustion engine and having an active element forcontacting an exhaust gas flowing through the exhaust passage and aheater for heating the active element, comprising: temperatureestimating means for sequentially estimating the temperature of theactive element based on a predetermined element temperature model whichis representative of, in combination, a temperature change of the activeelement due to heat transfer between said active element and an exhaustgas held in contact with the active element, and a temperature change ofthe active element due to heat transfer between the active element andsaid heater, and sequentially estimating the temperature of the heaterbased on a predetermined heater temperature model which isrepresentative of, in combination, a temperature change of the heaterdue to heat transfer between said heater and said active element and atemperature change of the heater due to the supply of heating energy tosaid heater; and heater control means for controlling said heater toequalize the temperature of the heater with a predetermined targettemperature, using an estimated value of the temperature of the activeelement and an estimated value of the temperature of the heater fromsaid temperature estimating means.
 34. The apparatus for controlling thetemperature of an exhaust gas sensor according to claim 33, wherein saidelement temperature model comprises a model which is representative of achange per predetermined time in the temperature of said active elementas including a temperature change component depending on the differencebetween the temperature of the active element and the temperature of theexhaust gas held in contact with the active element, a temperaturechange component depending on the difference between the temperature ofthe active element and the temperature of said heater; said heatertemperature model comprises a model which is representative of a changeper predetermined time in the temperature of said heater as including atemperature change component depending on the difference between thetemperature of the heater and the temperature of the active element, anda temperature change component depending on an amount of heating energysupplied to said heater; and said temperature estimating meanssequentially estimates a temperature change of said active element basedon said element temperature model, and accumulatively adds an estimatedvalue of the temperature change to an initial value of the temperatureof the active element which is set When said internal combustion enginestarts to operate, thereby estimating the temperature of the activeelement, and sequentially estimates a temperature change of said heaterbased on said heater temperature model, and accumulatively adds anestimated value of the temperature change to an initial value of thetemperature of the heater which is set when said internal combustionengine starts to operate, thereby estimating the temperature of theheater.
 35. The apparatus for controlling the temperature of an exhaustgas sensor according to claim 34, wherein said initial value is setdepending on the atmospheric temperature and/or the temperature of theinternal combustion engine at least when said internal combustion enginestarts to operate.
 36. The apparatus for controlling the temperature ofan exhaust gas sensor according to claim 33, wherein said heater controlmeans sequentially generates a control input which determines an amountof heating energy supplied to said heater by adding an input componentdepending on at least the estimated value of the temperature of theactive element from said temperature estimating means and an inputcomponent depending on at least the estimated value of the temperatureof the heater from said temperature estimating means, and controls saidheater depending on the control input.
 37. A recording medium readableby a computer and storing a temperature control program for enabling thecomputer to perform a process of controlling the temperature of anexhaust gas sensor disposed in an exhaust passage of.an internalcombustion engine and having an active element for contacting an exhaustgas flowing through the exhaust passage and a heater for heating theactive element, wherein: said temperature control program comprises atemperature estimating program for enabling said computer to perform aprocess of sequentially estimating the temperature of the active elementbased on a predetermined element temperature model which isrepresentative of a temperature change of the active element due to heattransfer between at least said active element and an exhaust gas held incontact with the active element, and a heater control program forenabling said computer to perform a process of controlling said heaterto equalize the temperature of the active element with a predeterminedtarget temperature, using an estimated value of the temperature of theactive element.
 38. The recording medium storing a temperature controlprogram for an exhaust gas sensor according to claim 37, wherein saidelement temperature model comprises a model which is determined torepresent, in combination, a temperature change of the active elementdue to heat transfer between said active element and an exhaust gas heldin contact with the active element and a temperature change of theactive element due to heat transfer between said active element and saidheater.
 39. The recording medium storing a temperature control programfor an exhaust gas sensor according to claim 38, wherein said elementtemperature model comprises a model which is representative of a changeper predetermined time in the temperature of said active element asincluding a temperature change component depending on the differencebetween the temperature of the active element and the temperature of theexhaust gas held in contact with the active element, and a temperaturechange component depending on the difference between the temperature ofthe active element and the temperature of said heater, and saidtemperature estimating program comprises a program for enabling thecomputer to perform a process of sequentially estimating a temperaturechange of said active element based on said element temperature model,and accumulatively adding an estimated value of the temperature changeto an initial value which is set when said internal combustion enginestarts to operate, thereby estimating the temperature of the activeelement.
 40. The recording medium storing a temperature control programfor an exhaust gas sensor according to claim 39, wherein said initialvalue is set depending on the atmospheric temperature andlor thetemperature of the internal combustion engine at least when saidinternal combustion engine starts to operate.
 41. The recording mediumstoring a temperature control program for an exhaust gas sensoraccording to claim 37, wherein said element temperature model comprisesa model which is representative of a change per predetermined time inthe temperature of said active element as including a temperature changecomponent depending on the difference between at least the temperatureof the active element and the temperature of the exhaust gas held incontact with the active element, and said temperature estimating programcomprises a program for enabling the computer to perform a process ofsequentially estimating a temperature change of said active elementbased on said element temperature model, and accumulatively adding anestimated value of the temperature change to an initial value which isset when said internal combustion engine starts to operate, therebyestimating the temperature of the active element.
 42. The recordingmedium storing a temperature control program for an exhaust gas sensoraccording to claim 41, wherein said initial value is set depending onthe atmospheric temperature and/or the temperature of the internalcombustion engine at least when said internal combustion engine startsto operate.
 43. The recording medium storing a temperature controlprogram for an exhaust gas sensor according to claim 37, wherein saidheater control program comprises a program for enabling said computer toperform a process of sequentially generating a control input whichdetermines an amount of heating energy supplied to said heater,depending on at least the estimated value of the temperature of theactive element, and controlling said heater depending on the controlinput.
 44. A recording medium readable by a computer and storing atemperature control program for enabling the computer to perform aprocess of controlling the temperature of an exhaust gas sensor disposedin an exhaust passage of an internal combustion engine and having anactive element for contacting an exhaust gas flowing through the exhaustpassage and a heater for heating the active element, wherein: saidtemperature control program comprises a temperature estimating programfor enabling said computer to perform a process of sequentiallyestimating the temperature of the active element based on apredetermined element temperature model which is representative of atemperature change of the active element due to heat transfer between atleast said active element and said heater, and a heater control programfor enabling said computer to perform a process of controlling saidheater to equalize the temperature of the active element with apredetermined target temperature, using an estimated value of thetemperature of the active element.
 45. The recording medium storing atemperature control program for an exhaust gas sensor according to claim44, wherein said element temperature model comprises a model which isrepresentative of a change per predetermined time in the temperature ofsaid active element as including a temperature change componentdepending on the difference between at least the temperature of theactive element and the temperature of said heater, and said temperatureestimating program comprises a program for enabling the computer toperform a process of sequentially estimating a temperature change ofsaid active element based on said element temperature model, andaccumulatively adding an estimated value of the temperature change to aninitial value which is set when said internal combustion engine startsto operate, thereby estimating the temperature of the active element.46. The recording medium storing a temperature control program for anexhaust gas sensor according to claim 45, wherein said initial value isset depending on the atmospheric temperature and/or the temperature ofthe internal combustion engine at least when said internal combustionengine starts to operate.
 47. The recording medium storing a temperaturecontrol program for an exhaust gas sensor according to claim 44, whereinsaid element temperature model comprises a model which is determined torepresent, in combination, a temperature change of the active elementdue to heat transfer between said active element and an exhaust gas heldin contact with the active element and a temperature change of theactive element due to heat transfer between said active element and saidheater.
 48. The recording medium storing a temperature control programfor an exhaust gas sensor according to claim 47, wherein said elementtemperature model comprises a model which is representative of a changeper predetermined time in the temperature of said active element asincluding a temperature change component depending on the differencebetween the temperature of the active element and the temperature of theexhaust gas held in contact with the active element, and a temperaturechange component depending on the difference between the temperature ofthe active element and the temperature of said heater, and saidtemperature estimating program comprises a program for enabling thecomputer to perform a process of sequentially estimating a temperaturechange of said active element based on said element temperature model,and accumulatively adding an estimated value of the temperature changeto an initial value which is set when said internal combustion enginestarts to operate, thereby estimating the temperature of the activeelement.
 49. The recording medium storing a temperature control programfor an exhaust gas sensor according to claim 48, wherein said initialvalue is set depending on the atmospheric temperature and/or thetemperature of the internal combustion engine at least when saidinternal combustion engine starts to operate.
 50. The recording mediumstoring a temperature control program for an exhaust gas sensoraccording to claim 44, wherein said heater control program comprises aprogram for enabling said computer to perform a process of sequentiallygenerating a control input which determines an amount of heating energysupplied to said heater, depending on at least the estimated value ofthe temperature of the active element, and controlling said heaterdepending on the control input.
 51. A recording medium readable by acomputer and storing a temperature control program for enabling thecomputer to perform a process of controlling the temperature of anexhaust gas sensor disposed in an exhaust passage of an internalcombustion engine and having an active element for contacting an exhaustgas flowing through the exhaust passage and a heater for heating theactive element, wherein: said temperature control program comprises atemperature estimating program for enabling said computer to perform aprocess of sequentially estimating the temperature of the heater basedon a predetermined heater temperature model which is representative of atemperature change of the heater due to heat transfer between at leastsaid heater and said active element, and a heater control program forenabling said computer to perform a process of controlling said heaterto equalize the temperature of the heater with a predetermined targettemperature, using an estimated value of the temperature of the heater.52. The recording medium storing a temperature control program for anexhaust gas sensor according to claim 51, wherein said heatertemperature model comprises a model which is determined to represent, incombination, a temperature change of the heater due to heat transferbetween said heater and said active element and a temperature change ofthe heater due to the supply of heating energy to said heater.
 53. Therecording medium storing a temperature control program for an exhaustgas sensor according to claim 52, wherein said heater temperature modelcomprises a model which is representative of a change per predeterminedtime in the temperature of said heater as including a temperature changecomponent depending on the difference between the temperature of theheater and the temperature of the active element, and a temperaturechange component depending on an amount of heating energy supplied tosaid heater, and said temperature estimating program comprises a programfor enabling the computer to perform a process of sequentiallyestimating a temperature change of said heater based on said heatertemperature model, and accumulatively adding an estimated value of thetemperature change to an initial value which is set when said internalcombustion engine starts to operate, thereby estimating the temperatureof the heater.
 54. The recording medium storing a temperature controlprogram for an exhaust gas sensor according to claim 53, wherein saidinitial value is set depending on the atmospheric temperature and/or thetemperature of the internal combustion engine at least when saidinternal combustion engine starts to operate.
 55. The recording mediumstoring a temperature control program for an exhaust gas sensoraccording to claim 51, wherein said heater temperature model comprises amodel which is representative of a change per predetermined time in thetemperature of said heater as including a temperature change componentdepending on the difference between the temperature of the heater andthe temperature of the active element, and said temperature estimatingprogram comprises a program for enabling the computer to perform aprocess of sequentially estimating a temperature change of said heaterbased on said heater temperature model, and accumulatively adding anestimated value of the temperature change to an initial value which isset when said internal combustion engine starts to operate, therebyestimating the temperature of the heater.
 56. The recording mediumstoring a temperature control program for an exhaust gas sensoraccording to claim 55, wherein said initial value is set depending onthe atmospheric temperature and/or the temperature of the internalcombustion engine at least when said internal combustion engine startsto operate.
 57. The recording medium storing a temperature controlprogram for an exhaust gas sensor according to claim 51, wherein saidheater control program comprises a program for enabling said computer toperform a process of sequentially generating a control input whichdetermines an amount of heating energy supplied to said heater,depending on at least the estimated value of the temperature of theheater, and controlling said heater depending on the control input. 58.A recording medium readable by a computer and storing a temperaturecontrol program for enabling the computer to perform a process ofcontrolling the temperature of an exhaust gas sensor disposed in anexhaust passage of an internal combustion engine and having an activeelement for contacting an exhaust gas flowing through the exhaustpassage and a heater for heating the active element, wherein: saidtemperature control program comprises a temperature estimating programfor enabling said computer to perform a process of sequentiallyestimating the temperature of the heater based on a predetermined heatertemperature model which is representative of a temperature change of theheater due to at least the supply of heating energy to said heater, anda heater control program for enabling said computer to perform a processof controlling said heater to equalize the temperature of the heaterwith a predetermined target temperature, using an estimated value of thetemperature of the heater.
 59. The recording medium storing atemperature control program for an exhaust gas sensor according to claim58, wherein said heater temperature model comprises a model which isrepresentative of a change per predetermined time in the temperature ofsaid heater as including a temperature change component depending on anamount of heating energy supplied to said heater, and said temperatureestimating program comprises a program for enabling the computer toperform a process of sequentially estimating a temperature change ofsaid heater based on said heater temperature model, and accumulativelyadds an estimated value of the temperature change to an initial valuewhich is set when said internal combustion engine starts to operate,thereby estimating the temperature of the heater.
 60. The recordingmedium storing a temperature control program for an exhaust gas sensoraccording to claim 59, wherein said initial value is set depending onthe atmospheric temperature and/or the temperature of the internalcombustion engine at least when said internal combustion engine startsto operate.
 61. The recording medium storing a temperature controlprogram for an exhaust gas sensor according to claim 58, wherein saidheater temperature model comprises a model which is determined torepresent, in combination, a temperature change of the heater due toheat transfer between said heater and said active element and atemperature change of the heater due to the supply of heating energy tosaid heater.
 62. The recording medium storing a temperature controlprogram for an exhaust gas sensor according to claim 61, wherein saidheater temperature model comprises a model which is representative of achange per predetermined time in the temperature of said heater asincluding a temperature change component depending on the differencebetween the temperature of the heater and the temperature of the activeelement, and a temperature change component depending on an amount ofheating energy supplied to said heater, and said temperature estimatingprogram comprises a program for enabling the computer to perform aprocess of sequentially estimating a temperature change of said heaterbased on said heater temperature model, and accumulatively adding anestimated value of the temperature change to an initial value which isset when said internal combustion engine starts to operate, therebyestimating the temperature of the heater.
 63. The recording mediumstoring a temperature control program for an exhaust gas sensoraccording to claim 62, wherein said initial value is set depending onthe atmospheric temperature and/or the temperature of the internalcombustion engine at least when said internal combustion engine startsto operate.
 64. The recording medium storing a temperature controlprogram for an exhaust gas sensor according to claim 58, wherein saidheater control program comprises a program for enabling said computer toperform a process of sequentially generating a control input whichdetermines an amount of heating energy supplied to said heater,depending on at least the estimated value of the temperature of theheater, and controlling said heater depending on the control input. 65.A recording medium readable by a computer and storing a temperaturecontrol program for enabling the computer to perform a process ofcontrolling the temperature of an exhaust gas sensor disposed in anexhaust passage of an internal combustion engine and having an activeelement for contacting an exhaust gas flowing through the exhaustpassage and a heater for heating the active element, wherein: saidtemperature control program comprises a temperature estimating programfor enabling said computer to perform a process of sequentiallyestimating the temperature of the active element based on apredetermined element temperature model which is representative of, incombination, a temperature change of the active element due to heattransfer between said active element and an exhaust gas held in contactwith the active element, and a temperature change of the active elementdue to heat transfer between the active element and said heater, andsequentially estimating the temperature of the heater based on apredetermined heater temperature model which is representative of, incombination, a temperature change of the heater due to heat transferbetween said heater and said active element and a temperature change ofthe heater due to the supply of heating energy to said heater; and aheater control program for enabling said computer to perform a processof controlling said heater to equalize the temperature of the activeelement with a predetermined target temperature, using an estimatedvalue of the temperature of the active element and an estimated value ofthe temperature of the heater.
 66. The recording medium storing atemperature control program for an exhaust gas sensor according to claim65, wherein said element temperature model comprises a model which isrepresentative of a change per predetermined time in the temperature ofsaid active element as including a temperature change componentdepending on the difference between the temperature of the activeelement and the temperature of the exhaust gas held in contact with theactive element, a temperature change component depending on thedifference between the temperature of the active element and thetemperature of said heater; said heater temperature model comprises amodel which is representative of a change per predetermined time in thetemperature of said heater as including a temperature change componentdepending on the difference between the temperature of the heater andthe temperature of the active element, and a temperature changecomponent depending on an amount of heating energy supplied to saidheater; and said temperature estimating program comprises a program forenabling the computer to perform a process of sequentially estimating atemperature change of said active element based on said elementtemperature model, and accumulatively adding an estimated value of thetemperature change to an initial value of the temperature of the activeelement which is set when said internal combustion engine starts tooperate, thereby estimating the temperature of the active element, andsequentially estimating a temperattire change of said heater based onsaid heater temperature model, and accumulatively adding an estimatedvalue of the temperature change to an initial value of the temperatureof the heater which is set when said internal combustion engine startsto operate, thereby estimating the temperature of the heater.
 67. Therecording medium storing a temperature control program for an exhaustgas sensor according to claim 66, wherein said initial value is setdepending on the atmospheric temperature and/or the temperature of theinternal combustion engine at least when said internal combustion enginestarts to operate.
 68. The recording medium storing a temperaturecontrol program for an exhaust gas sensor according to claim 65, whereinsaid heater control program comprises a program for enabling saidcomputer to perform a process of sequentially generating a control inputwhich determines an amount of heating energy supplied to said heater byadding an input component depending on at least the estimated value ofthe temperature of the active element and the estimated value of thetemperature of the heater, and controlling said heater depending on thecontrol input.
 69. A recording medium readable by a computer and storinga temperature control program for enabling the computer to perform aprocess of controlling the temperature of an exhaust gas sensor disposedin an exhaust passage of an internal combustion engine and having anactive element for contacting an exhaust gas flowing through the exhaustpassage and a heater for heating the active element, wherein: saidtemperature control program comprises a temperature estimating programfor enabling said computer to perform a process of sequentiallyestimating the temperature of the active element based on apredetermined element temperature model which is representative of, incombination, a temperature change of the active element due to heattransfer between said active element and an exhaust gas held in contactwith the active element, and a temperature change of the active elementdue to heat transfer between the active element and said heater, andsequentially estimating the temperature of the heater based on apredetermined heater temperature model which is representative of, incombination, a temperature change of the heater due to heat transferbetween said heater and said active element and a temperature change ofthe heater due to the supply of heating energy to said heater; and aheater control program for enabling said computer to perform a processof controlling said heater to equalize the temperature of the heaterwith a predetermined target temperature, using an estimated value of thetemperature of the active element and an estimated value of thetemperature of the heater.
 70. The recording medium storing atemperature control program for an exhaust gas sensor according to claim69, wherein said element temperature model comprises a model which isrepresentative of a change per predetermined time in the temperature ofsaid active element as including a temperature change componentdepending on the difference between the temperature of the activeelement and the temperature of the exhaust gas held in contact with theactive element, a temperature change component depending on thedifference between the temperature of the active element and thetemperature of said heater; said heater temperature model comprises amodel which is representative of a change per predetermined time in thetemperature of said heater as including a temperature change componentdepending on the difference between the temperature of the heater andthe temperature of the active element, and a temperature changecomponent depending on an amount of heating energy supplied to saidheater; and said temperature estimating program comprises a program forenabling the computer to perform a process of sequentially estimating atemperature change of said active element based on said elementtemperature model, and accumulatively adding an estimated value of thetemperature change to an initial value of the temperature of the activeelement which is set when said internal combustion engine starts tooperate, thereby estimating the temperature of the active element, andsequentially estimating a temperature change of said heater based onsaid heater temperature model, and accumulatively adding an estimatedvalue of the temperature change to an initial value of the temperatureof the heater which is set when said internal combustion engine startsto operate, thereby estimating the temperature of the heater.
 71. Therecording medium storing a temperature control program for an exhaustgas sensor according to claim 70, wherein said initial value is setdepending on the atmospheric temperature and/or the temperature of theinternal combustion engine at least when said internal combustion enginestarts to operate.
 72. The recording medium storing a temperaturecontrol program for an exhaust gas sensor according to claim 69, whereinsaid heater control program comprises a program for enabling saidcomputer to perform a process of sequentially generating a control inputwhich determines an amount of heating energy supplied to said heater byadding an input component depending on at least the estimated value ofthe temperature of the active element and the estimated value of thetemperature of the heater, and controlling said heater depending on thecontrol input.
 73. A method of controlling the temperature of an exhaustgas sensor disposed in an exhaust passage of an internal combustionengine and having an active element for contacting an exhaust gasflowing through the exhaust passage and a heater for heating the activeelement, comprising: while sequentially estimating the temperature ofthe active element based on a predetermined element temperature modelwhich is representative of a temperature change of the active elementdue to heat transfer between at least said active element and an exhaustgas held in contact with the active element, controlling said heater toequalize the temperature of the active element with a predeterminedtarget temperature, using an estimated value of the temperature of theactive element.
 74. The method of controlling the temperature of anexhaust gas sensor according to claim 73, wherein said elementtemperature model comprises a model which is determined to represent, incombination, a temperature change of the active element due to heattransfer between said active element and an exhaust gas held in contactwith the active element and a temperature change of the active elementdue to heat transfer between said active element and said heater. 75.The method of controlling the temperature of an exhaust gas sensoraccording to claim 74, wherein said element temperature model comprisesa model which is representative of a change per predetermined time inthe temperature of said active element as including a temperature changecomponent depending on the difference between the temperature of theactive element and the temperature of the exhaust gas held in contactwith the active element, and a temperature change component depending onthe difference between the temperature of the active element and thetemperature of said heater, and while sequentially estimating atemperature change of said active element based on said elementtemperature model, an estimated value of the temperature change isaccumulatively added to an initial value which is set when said internalcombustion engine starts to operate, thereby estimating the temperatureof the active element.
 76. The method of controlling the temperature ofan exhaust gas sensor according to claim 75, wherein said initial valueis set depending on the atmospheric temperature and/or the temperatureof the internal combustion engine at least when said internal combustionengine starts to operate.
 77. The method of controlling the temperatureof an exhaust gas sensor according to claim 73, wherein said elementtemperature model comprises a model which is representative of a changeper predetermined time in the temperature of said active element asincluding a temperature change component depending on the differencebetween at least the temperature of the active element and thetemperature of the exhaust gas held in contact with the active element,and while sequentially estimating a temperature change of said activeelement based on said element temperature model, an estimated value ofthe temperature change is accumulatively added to an initial value whichis set when said internal combustion engine starts to operate, therebyestimating the temperature of the active element.
 78. The method ofcontrolling the temperature of an exhaust gas sensor according to claim77, wherein said initial value is set depending on the atmospherictemperature and/or the temperature of the internal combustion engine atleast when said internal combustion engine starts to operate.
 79. Themethod of controlling the temperature of an exhaust gas sensor accordingto claim 73, wherein while sequentially generating a control input whichdetermines an amount of heating energy supplied to said heater,depending on at least the estimated value of the temperature of theactive element, said heater is controlled depending on the controlinput.
 80. A method of controlling the temperature of an exhaust gassensor disposed in an exhaust passage of an internal combustion engineand having an active element for contacting an exhaust gas flowingthrough the exhaust passage and a heater for heating the active element,comprising: while sequentially estimating the temperature of the activeelement based on a predetermined element temperature model which isrepresentative of a temperature change of the active element due to heattransfer between at least said active element and said heater,controlling said heater to equalize the temperature of the activeelement with a predetermined target temperature, using an estimatedvalue of the temperature of the active element.
 81. The method ofcontrolling the temperature of an exhaust gas sensor according to claim80, wherein said element temperature model comprises a model which isrepresentative of a change per predetermined time in the temperature ofsaid active element as including a temperature change componentdepending on the difference between at least the temperature of theactive element and the temperature of said heater, and whilesequentially estimating a temperature change of said active elementbased on said element temperature model, an estimated value of thetemperature change is accumulatively added to an initial value which isset when said internal combustion engine starts to operate, therebyestimating the temperature of the active element.
 82. The method ofcontrolling the temperature of an exhaust gas sensor according to claim81, wherein said initial value is set depending on the atmospherictemperature and/or the temperature of the internal combustion engine atleast when said internal combustion engine starts to operate.
 83. Themethod of controlling the temperature of an exhaust gas sensor accordingto claim 80, wherein said element temperature model comprises a modelwhich is determined to represent, in combination, a temperature changeof the active element due to heat transfer between said active elementand an exhaust gas held in contact with the active element and atemperature change of the active element due to heat transfer betweensaid active element and said heater.
 84. The method of controlling thetemperature of an exhaust gas sensor according to claim 83, wherein saidelement temperature model comprises a model which is representative of achange per predetermined time in the temperature of said active elementas including a temperature change component depending on the differencebetween the temperature of the active element and the temperature of theexhaust gas held in contact with the active element, and a temperaturechange component depending on the difference between the temperature ofthe active element and the temperature of said heater, and whilesequentially estimating a temperature change of said active elementbased on said element temperature model, an estimated value of thetemperature change is accumulatively added to an initial value which isset when said internal combustion engine starts to operate, therebyestimating the temperature of the active element.
 85. The method ofcontrolling the temperature of an exhaust gas sensor according to claim84, wherein said initial value is set depending on the atmospherictemperature and/or the temperature of the internal combustion engine atleast when said internal combustion engine starts to operate.
 86. Themethod of controlling the temperature of an exhaust gas sensor accordingto claim 80, wherein while sequentially generating a control input whichdetermines an amount of heating energy supplied to said heater,depending on at least the estimated value of the temperature of theactive element, said heater is controlled depending on the controlinput.
 87. A method of controlling the temperature of an exhaust gassensor disposed in an exhaust passage of an internal combustion engineand having an active element for contacting an exhaust gas flowingthrough the exhaust passage and a heater for heating the active element,comprising: while sequentially estimating the temperature of the heaterbased on a predetermined heater temperature model which isrepresentative of a temperature change of the heater due to heattransfer between at least said heater and said active element,controlling said heater to equalize the temperature of the heater with apredetermined target temperature, using an estimated value of thetemperature of the heater.
 88. The method of controlling the temperatureof an exhaust gas sensor according to claim 87, wherein said heatertemperature model comprises a model which is determined to represent, incombination, a temperature change of the heater due to heat transferbetween said heater and said active element and a temperature change ofthe heater due to the supply of heating energy to said heater.
 89. Themethod of controlling the temperature of an exhaust gas sensor accordingto claim 88, wherein said heater temperature model comprises a modelwhich is representative of a change per predetermined time in thetemperature of said heater as including a temperature change componentdepending on the difference between the temperature of the heater andthe temperature of the active element, and a temperature changecomponent depending on an amount of heating energy supplied to saidheater, and while sequentially estimating a temperature change of saidheater based on said heater temperature model, an estimated value of thetemperature change is accumulatively added to an initial value which isset when said internal combustion engine starts to operate, therebyestimating the temperature of the heater.
 90. The method of controllingthe temperature of an exhaust gas sensor according to claim 89, whereinsaid initial value is set depending on the atmospheric temperatureand/or the temperature of the internal combustion engine at least whensaid internal combustion engine starts to operate.
 91. The method ofcontrolling the temperature of an exhaust gas sensor according to claim87, wherein said heater temperature model comprises a model which isrepresentative of a change per predetermined time in the temperature ofsaid heater as including a temperature change component depending on thedifference between the temperature of the heater and the temperature ofthe active element, and while sequentially estimating a temperaturechange of said heater based on said heater temperature model, anestimated value of the temperature change is accumulatively added to aninitial value which is set when said internal combustion engine startsto operate, thereby estimating the temperature of the heater.
 92. Themethod of controlling the temperature of an exhaust gas sensor accordingto claim 91, wherein said initial value is set depending on theatmospheric temperature and/or the temperature of the internalcombustion engine at least when said internal combustion engine startsto operate.
 93. The method of controlling the temperature of an exhaustgas sensor according to claim 87, wherein while sequentially generatinga control input which determines an amount of heating energy supplied tosaid heater, depending on at least the estimated value of thetemperature of the heater, said heater is controlled depending on thecontrol input.
 94. A method of controlling the temperature of an exhaustgas sensor disposed in an exhaust passage of an internal combustionengine and having an active element for contacting an exhaust gasflowing through the exhaust passage and a heater for heating the activeelement, comprising: while sequentially estimating the temperature ofthe heater based on a predetermined heater temperature model which isrepresentative of a temperature change of the heater due to the supplyof heating energy to at least said heater, controlling said heater toequalize the temperature of the heater with a predetermined targettemperature, using an estimated value of the temperature of the heater.95. The method of controlling the temperature of an exhaust gas sensoraccording to claim 94, wherein said heater temperature model comprises amodel which is representative of a change per predetermined time in thetemperature of said heater as including a temperature change componentdepending on an amount of heating energy supplied to said heater, andwhile sequentially estimating a temperature change of said heater basedon said heater temperature model, an estimated value of the temperaturechange is accumulatively added to an initial value which is set whensaid internal combustion engine starts to operate, thereby estimatingthe temperature of the heater.
 96. The method of controlling thetemperature of an exhaust gas sensor according to claim 95, wherein saidinitial value is set depending on the atmospheric temperature and/or thetemperature of the internal combustion engine at least when saidinternal combustion engine starts to operate.
 97. The method ofcontrolling the temperature of an exhaust gas sensor according to claim94, wherein said heater temperature model comprises a model which isdetermined to represent, in combination, a temperature change of theheater due to heat transfer between said heater and said active elementand a temperature change of the heater due to the supply of heatingenergy to said heater.
 98. The method of controlling the temperature ofan exhaust gas sensor according to claim 97, wherein said heatertemperature model comprises a model which is representative of a changeper predetermined time in the temperature of said heater as including atemperature change component depending on the difference between thetemperature of the heater and the temperature of the active element, anda temperature change component depending on an amount of heating energysupplied to said heater, and while sequentially estimating a temperaturechange of said heater based on said heater temperature model, anestimated value of the temperature change is accumulatively added to aninitial value which is set when said internal combustion engine startsto operate, thereby estimating the temperature of the heater.
 99. Themethod of controlling the temperature of an exhaust gas sensor accordingto claim 98, wherein said initial value is set depending on theatmospheric temperature and/or the temperature of the internalcombustion engine at least when said internal combustion engine startsto operate.
 100. The method of controlling the temperature of an exhaustgas sensor according to claim 94, wherein while sequentially generatinga control input which determines an amount of heating energy supplied tosaid heater, depending on at least the estimated value of thetemperature of the heater, said heater is controlled depending on thecontrol input.
 101. A method of controlling the temperature of anexhaust gas sensor disposed in an exhaust passage of an internalcombustion engine and having an active element for contacting an exhaustgas flowing through the exhaust passage and a heater for heating theactive element, comprising: sequentially estimating the temperature ofthe active element based on a predetermined element temperature modelwhich is representative of, in combination, a temperature change of theactive element due to heat transfer between said active element and anexhaust gas held in contact with the active element, and a temperaturechange of the active element due to heat transfer between the activeelement and said heater, and sequentially estimating the temperature ofthe heater based on a predetermined heater temperature model which isrepresentative of, in combination, a temperature change of the heaterdue to heat transfer between said heater and said active element and atemperature change of the heater due to the supply of heating energy tosaid heater, and controlling said heater to equalize the temperature ofthe active element with a predetermined target temperature, using anestimated value of the temperature of the active element and anestimated value of the temperature of the heater while estimating thetemperature of the active element and the temperature of the heater.102. The method of controlling the temperature of an exhaust gas sensoraccording to claim 101, wherein said element temperature model comprisesa model which is representative of a change per predetermined time inthe temperature of said active element as including a temperature changecomponent depending on the difference between the temperature of theactive element and the temperature of the exhaust gas held in contactwith the active element, a temperature change component depending on thedifference between the temperature of the active element and thetemperature of said heater; said heater temperature model comprises amodel which is representative of a change per predetermined time in thetemperature of said heater as including a temperature change componentdepending on the difference between the temperature of the heater andthe temperature of the active element, and a temperature changecomponent depending on an amount of heating energy supplied to saidheater; and while sequentially estimating a temperature change of saidactive element based on said element temperature model, an estimatedvalue of the temperature change is accumulatively added to an initialvalue of the temperature of the active element which is set when saidinternal combustion engine starts to operate, thereby estimating thetemperature of the active element, and while sequentially estimating atemperature change of said heater based on said heater temperaturemodel, an estimated value of the temperature change is accumulativelyadded to an initial value of the temperature of the heater which is setwhen said internal combustion engine starts to operate, therebyestimating the temperature of the heater.
 103. The method of controllingthe temperature of an exhaust gas sensor according to claim 102, whereinsaid initial value is set depending on the atmospheric temperatureand/or the temperature of the internal combustion engine at least whensaid internal combustion engine starts to operate.
 104. The method ofcontrolling the temperature of an exhaust gas sensor according to claim101, wherein while sequentially generating a control input whichdetermines an amount of heating energy supplied to said heater by addingan input component depending on at least the estimated value of thetemperature of the active element and an input component depending on atleast the estimated value of the temperature of the heater, said heateris controlled depending on the control input.
 105. A method ofcontrolling the temperature of an exhaust gas sensor disposed in anexhaust passage of an internal combustion engine and having an activeelement for contacting an exhaust gas flowing through the exhaustpassage and a heater for heating the active element, comprising:sequentially estimating the temperature of the active element based on apredetermined element temperature model which is representative of, incombination, a temperature change of the active element due to heattransfer between said active element and an exhaust gas held in contactwith the active element, and a temperature change of the active elementdue to heat transfer between the active element and said heater, andsequentially estimating the temperature of the heater based on apredetermined heater temperature model which is representative of, incombination, a temperature change of the heater due to heat transferbetween said heater and said active element and a temperature change ofthe heater due to the supply of heating energy to said heater, andcontrolling said heater to equalize the temperature of the heater with apredetermined target temperature, using an estimated value of thetemperature of the active element and an estimated value of thetemperature of the heater while estimating the temperature of the activeelement and the temperature of the heater.
 106. The method ofcontrolling the temperature of an exhaust gas sensor according to claim100, wherein said element temperature model comprises a model which isrepresentative of a change per predetermined time in the temperature ofsaid active element as including a temperature change componentdepending on the difference between the temperature of the activeelement and the temperature of the exhaust gas held in contact with theactive element, a temperature change component depending on thedifference between the temperature of the active element and thetemperature of said heater; said heater temperature model comprises amodel which is representative of a change per predetermined time in thetemperature of said heater as including a temperature change componentdepending on the difference between the temperature of the heater andthe temperature of the active element, and a temperature changecomponent depending on an amount of heating energy supplied to saidheater; and while sequentially estimating a temperature change of saidactive element based on said element temperature model, an estimatedvalue of the temperature change is accumulatively added to an initialvalue of the temperature of the active element which is set when saidinternal combustion engine starts to operate, thereby estimating thetemperature of the active element, and while sequentially estimating atemperature change of said heater based on said heater temperaturemodel, an estimated value of the temperature change is accumulativelyadded to an initial value of the temperature of the heater which is setwhen said internal combustion engine starts to operate, therebyestimating the temperature of the heater.
 107. The method of controllingthe temperature of an exhaust gas sensor according to claim 106, whereinsaid initial value is set depending on the atmospheric temperatureand/or the temperature of the internal combustion engine at least whensaid internal combustion engine starts to operate.
 108. The method ofcontrolling the temperature of an exhaust gas sensor according to claim105, wherein while sequentially generating a control input whichdetermines an amount of heating energy supplied to said heater by addingan input component depending on at least the estimated value of thetemperature of the active element and an input component depending on atleast the estimated value of the temperature of the heater, said heateris controlled depending on the control input.